There were certainly instances of what could be romantically described as “chivalry” in the air in WWI. Perhaps the best documented is Udet vs. Guynemer. Here is a video that dramatizes the clash. To summarize, the two planes met and began to turn into each other. Udet’s gun jammed, leaving him quite vulnerable. As he tried to clear the jam, he saw that Guynemer noticed his problem. Given that Guynemer was an excellent pilot and no longer had to worry about being shot in return, he could have easily taken more risks and shot Udet out of the sky. Instead, on the next pass, Guynemer waved rather than shoot his adversary.

This wasn’t an isolated incident. Many of the aerial units were previously cavalry units and there was some carryover from that culture. Also, there was a good bit of romance in the early war. Pilots would party all night (partly to fight the extreme stress of being a combat pilot), and would sometimes show up for the “dawn patrol” still in their tuxedo and hungover from the night before. As the war went along, greater numbers and increased hatred for the enemy reduced the number of chivalrous incidents in the air. A simple way of looking at it is that when there are more planes in the air on both sides, you simply cannot feel as safe as Guynemer did in his one-on-one dogfight vs. Udet. Having your friends killed by the enemy also often will inflame your feelings against them, reducing any chivalrous impulses.

Putting romance and sporadic incidents aside, the real “code” of the air war in WWI was the Dicta Boelke (and Allied counterparts from pilots like Edward Mannock and others). This set of rules for aerial combat was compiled by Oswald Boelke. Wikipedia’s list will suffice.

1. Try to secure the upper hand before attacking. If possible, keep the sun behind you
2. Always continue with an attack you have begun
3. Open fire only at close range, and then only when the opponent is squarely in your sights
4. You should always try to keep your eye on your opponent, and never let yourself be deceived by ruses
5. In any type of attack, it is essential to assail your opponent from behind
6. If your opponent dives on you, do not try to get around his attack, but fly to meet it
7. When over the enemy's lines, always remember your own line of retreat
8. Tip for Squadrons: In principle, it is better to attack in groups of four or six. Avoid two aircraft attacking the same opponent

You will notice nothing chivalrous—this list is all business. If you want to kill the enemy, this is what you were supposed to do. If your enemy’s gun had jammed like Udet’s, all the better for you. These concepts are still taught today, which is a testament to how well the business of killing your opponents in the air was understood despite its novelty. Despite the massive changes between a Fokker Eindecker and an F-22 Raptor, these ideas apply to an astonishing degree (though some need to be reinterpreted to allow for stealth, missiles, radar, and other modern equipment). Any chivalric code among WWI aviators was rapidly outstripped by lists like this and the ideals they propose: kill your enemy as efficiently as possible and try to stay alive.

As usual, /u/Domini_canes has covered the matter pretty well.

One thing that I think you could say is that while pilots had a code, overall it was more of a mutual respect for one another, but still an understanding that ultimately they were at war.

There are many reports of pilots treating each other with respect, and it wasn't uncommon for both sides to exchange messages when possible by dropping little canisters filled with notes or other such means. For example, when a noted pilot was shot down behind enemy lines, it wasn't uncommon for the side that shot him down to drop a note informing the other side as to their condition (whether they were dead, or injured and captured, etc). Also, when a noted pilot was shot down and killed behind enemy lines, it was rather common for them to be buried with full military honors (or at least, as much as is possible during wartime). For example, Manfred von Richthofen, the famed Red Baron, was buried with honor by the the personnel of No. 3 Squadron, Australian Flying Corps. Squadrons stationed near the area came to pay their respects, with some leaving leaves. One such wreath was famously inscribed with "To our gallant and worthy foe". Some in the press, and scholars later, have noted that he received more of a funeral than many British aces received from their own side.

Now, on the topic of Richthofen himself, and to address the issue of how the era is often romanticized, despite what a recent movie may have you believe, he had absolutely no qualms about aiming for the crew of an aircraft. In the movie "The Red Baron", he tells his other pilots something along the lines of "we are not butchers! We aim for the aircraft, not the man!" Well, the exact opposite true. Richthofen, who subscribed to /u/Domini_canes' aforementioned Dicta Boelcke, advocated and told the men he served with, specifically to aim for the air crews of their targets. In the case of multiple crew aircraft with defensive armament, he advocated for the killing of the defensive gunners first and foremost. Once they were taken care of, or if they were not present in the target at hand, one was to aim specifically for the pilot. Taking out the pilot is a guaranteed way of bringing down the aircraft, whereas damage to the aircraft itself may still leave it flyable, and able to return home.

On the subject of the Dicta Boelcke, I think it perfectly illustrates just how pragmatic the pilots were, and how much they understood they were still ultimately at war. Rene Fonck, the leading Allied ace of the war, also followed similar such rules regarding air combat. They weren't looking for a fair or sporting fight. They wanted to use every advantage they could.

A leading ace for the British had his own set of rules, which are mostly similar to the Dicta Boelcke, but also has some differences - though it should be noted that just because Boelcke might have omitted some of the rules found on Mannock's list, he almost certainly would have approved of them and would have already incorporated them into his own flying in his own way. Portions in parenthesis are my additions explaining/talking about the rules.

1. Pilots must dive to attack with zest, and must hold their fire until they get within one hundred yards of their target. (This is common advice you will hear from expert pilots, and many credit their success to close range shots. Not only does it conserve ammunition and make the rounds that do hit more effective by virtue of having lost less velocity, it allows you to engage more targets, and it can limit the time you spend behind an enemy aircraft, wherein you might be vulnerable to attack - especially if you fall into the trap of "target fixation" which is where your focus is solely on your target. One would be vulnerable not just to other aircraft but defensive armament of the target as well.)

2. Achieve surprise by approaching from the East. (The East being the German side of the front. In other words, they'll be expecting you to come from your own side, so try and sneak around and come back to hit them from behind.)

3. Use the sun's glare and clouds to achieve surprise.

4. Pilots must keep physically fit by exercise and the moderate use of stimulants. (This is one of the earlier instances I've seen stressing the importance of physical fitness as being important to a pilot before. Again, it's just one of the earliest instances I'm aware of, not the earliest and it wasn't a totally revolutionary idea. You have to remember that for the vast majority, these aircraft were controlled directly by the pilot via cables and pulleys. So not only did a pilot have to contend with g forces which can be extremely taxing, but it sometimes required great strength to work the controls of an aircraft in combat. The use of amphetamines/stimulants by pilots survives to this day even.)

5. Pilots must sight their guns and practice as much as possible as targets are normally fleeting.

6. Pilots must practice spotting machines in the air and recognizing them at long range, and every airplane is to be treated as an enemy until it is certain it is not.

7. Pilots must learn where the enemy's blind spots are.

8. Scouts must be attacked from above and two-seaters from beneath their tails. (Scouts would generally be single engine, single crew aircraft with forward firing machine guns. Attacking from above allows you to dive on them - you're trading altitude for airspeed, which is a fundamental idea in air combat maneuvering. You can then trade that airspeed you gained in the dive for more altitude if need be. Two-seaters can in this case be interpreted as basically any aircraft with a pilot and an observer/defensive gunner. While the defensive guns might have quite a wide field of fire due to being mounted on a rotating turret, they will almost universally have a blind spot underneath the aircraft where they can't aim.)

9. Pilots must practice quick turns, as this maneuver is more used than any other in a fight.

10. Pilots must practice judging distances in the air as these are very deceptive.

11. Decoys should be guarded against. A single enemy is often a decoy, therefore the air above should be searched before attacking.

12. If the day is sunny, machines should be turned with as little bank as possible, otherwise the sun glistening on the wings will give away their presence at a long range.

13. Pilots must keep turning in a dog fight and never fly straight except when firing.

14. Pilots must never, under any circumstances, dive away from an enemy, as he gives his opponent a non-deflection shot — bullets are faster than airplanes.

15. Pilots must keep their eye on their watches during patrols, and on the direction and strength of the wind. (To give an idea of how this seemingly mundane aspect can actually be a serious issue, some blame the wind as being part of the reason for Richthofen's death. It is speculated that the wind was uncharacteristically blowing at ~25mph easterly, as opposed to the usual prevailing westerly direction. It has been speculated that the wind, acting as a tailwind, helped to more quickly push him over enemy lines. If true, this could partially explain why he violated his own rules against flying low over enemy territory, and how he was able to be brought down by ground fire (while his death is officially credited to ace Canadian pilot Arthur Roy Brown, I subscribe to the theory he was brought down by fire from Cedric Popkin's Vickers machine gun)

And while they were often stone cold killers, they weren't all without remorse. Mick Mannock, as his diary and conversations with other pilots report, was deeply affected by the death and destruction around him, and that which he was the cause of. As a side note, the awful movie Flyboys, which is rife with historical inaccuracies, does sort of in a way reference him. He was in particular afraid of his aircraft catching fire and dying in the flames. In the movie, there is a scene where a veteran pilot hands out revolvers to the new recruits before their first mission, saying (to paraphrase) "If your plane catches, you have three options. You can burn with it, you can jump out from 10,000 feet, or you can take the easy way out." (Also note that in the war, commanders were hesitant to give out parachutes, for fear that a pilot would be too quick to abandon his airplane instead of trying to get it home. There was also among some pilots a sense that a parachute conferred cowardice, and still others felt they were worthless and added unnecessary weight since they were indeed in their infancy and not very reliable.) Now, it should be noted that this "shoot yourself instead of burning" wasn't standard practice or a doctrine promoted through official training. As he noted to another pilot, the other pilots made fun of him for carrying the revolver. Not because he intended to use it against himself should his aircraft catch fire, but because they thought he intended to use it as an offensive weapon. To paraphrase (I don't have the source available at the moment), he says ".... but they're wrong (in regards to them thinking it's a weapon for use against the enemy). I bring it for the first signs of flames to finish myself off. They (the flames) will never burn me." This fear of burning in the aircraft came after he had witnessed his victims being subjected to the same fate. Unfortunately for Mannock, his final flight ended when his engine caught fire, and he crashed. It is not known whether his body was ever actually recovered, but the body that was supposedly his did not have a bullet wound in the head (or any self inflicted wound). That doesn't rule out that the body was indeed Mannock's of course, as there could be any number of reasons he was unable to "finish himself off before the flames did him in."