There was a special honour for doing just that in Ancient Rome called the spolia opima.

The tradition was supposed to have started with Romulus, who was said to have killed king Acron of the Caeninites and stripped him of his armour. He brought this back to Rome and dedicated it to Jupiter. Here's Livy 1.10:

He was no less anxious to display his achievements than he had been great in performing them, so, after leading his victorious army home, he mounted to the Capitol with the spoils of his dead foe borne before him on a frame constructed for the purpose. He hung them there on an oak, which the shepherds looked upon as a sacred tree, and at the same time marked out the site for the temple of Jupiter, and addressing the god by a new title, uttered the following invocation: ‘Jupiter Feretrius! these arms taken from a king, I, Romulus a king and conqueror, bring to thee, and on this domain, whose bounds I have in will and purpose traced, I dedicate a temple to receive the spolia opima which posterity following my example shall bear hither, taken from the kings and generals of our foes slain in battle.’

Such was the origin of the first temple dedicated in Rome. And the gods decreed that though its founder did not utter idle words in declaring that posterity would thither bear their spoils, still the splendour of that offering should not be dimmed by the number of those who have rivalled his achievement. For after so many years have elapsed and so many wars been waged, only twice have the spolia opima been offered. So seldom has Fortune granted that glory to men.

One thing Livy notes here is that is was a rare occurrence; the situation wasn't likely to come about often. But he did write about two other examples.

As a tribune in 437 BCE, Aulus Cornelius Cossus killed Lars Tolumnius, the king of Veii, took his armour from his body, and brought it back to dedicate it to Jupiter Feretrius. While he was apparently the highlight of the subsequent Triumph, as a tribune, Cossus was not technically able to offer spolia opima at the time. However, he became consul seven years later and apparently when Augustus went to view the armour he noted an inscription reading Aulus Cornelius Cossus, consul (Livy, 4:20). Augustus might have been messing around here, though, as he wanted to block M. Crassus' honour of dedicating the armour of the king of the Bastarnae of 29 BCE, which I'll get to in a moment.

The second historical case Livy mentions is that of Marcus Claudius Marcellus, consul in 222 BCE, who went on campaign against the Insubrians, a group of Gauls then active in Etruria. M. Marcellus killed Viridomarus, the chief of the Insubrians, and took his arms back to the city in the Triumph.

As I mentioned, Marcus Licinius Crassus, who was consul in 30 BCE (appointed by Octavian, later known as Augustus), campaigned successfully as a proconsul against the Bastarnae in Thrace and Macedonia. He fought their king, Deldo, in single combat, and after killing him he stripped the body of its armour. Adding to the glory of his victory was the fact that he also recaptured the legionary standards that the Bastarnae had captured from G. Antonius (Mark Antony's brother!) in 43 BCE. He was hailed as imperator; the Senate voted to grant M. Crassus a Triumph ex Thraecia et Geteis for July, 27 BCE.

Until this time the requirements for dedicating spolia opima seem to have been rather vague. Festus wrote:

M. Varro says that the spolia opima can be such, even if a common soldier has taken them, provided it is from an enemy commander.

and Plutarch also gave an account in which the spolia opima of a general killing a general was only one kind of many (Marcellus, 8). However, allowing another general to hold imperium and dedicate the spolia opima during Octavian's time would have been unacceptable. Thus Octavian may have forged a requirement that the dedicator had to be a consul, probably after making a private agreement with M. Crassus, thereby preventing a political crisis and challenge to his legitimacy.

Elsewhere in the ancient world, Ashurbanipal claimed to have killed king Teumman of Elam with his own sword, though this was probably propaganda and he was not really present on the campaign. I can't think of any other Mesopotamian examples off the top of my head. Mursili I was killed by his brother-in-law Hantili I and his son-in-law Zidanta I but we don't know how or who dealt the blows.

Yes, it happened in Bolivia in the middle of the nineteenth century, when one caudillo, Mariano Melgarejo, personally shot dead his predecessor, Manuel Belzú, and displayed his body from the presidential balcony in La Paz.

I described the background to these remarkable events in a longer essay that examined the suggestion that Queen Victoria had been so angered by Belzú's earlier treatment of her ambassador that she, symbolically, struck Bolivia off her map of South America, before ceasing formal diplomatic relations between the countries for a couple of generations. Exactly what happened between the two generals is somewhat disputed, but the most widely accepted version of events, offered by Paul Lewis, suggests that, in 1864, Belzú and Melgarejo became rivals plotting the overthrow of the highly unpopular President José María Achá.

According to Lewis, Belzú seized La Paz while his opponent was engaged in fighting in the countryside. Recognizing that he had little chance of forcing his way into the city, Melgarejo instead tried subterfuge, having two of his men escort him to the presidential palace as a “prisoner.” Belzú, who had been delivering one of his fiery speeches to a large crowd, was alerted to this astounding piece of good fortune and hurried over to gloat, whereupon Melgarejo drew a pistol, shot his rival dead and hauled his body up to the balcony from which Belzú had been addressing his supporters. Displaying the bloody corpse to the crowd, Melgarejo demanded “¿Belzú o Melgarejo?” And, after a few seconds of stunned silence, the crowd roared back: “¡Viva Melgarejo!”

Sources

Robert Barton. A Short History of the Republic of Bolivia. La Paz: Editorial Los Amigos del Libro, 1968

Paul Lewis. Authoritarian Leaders in Latin America: Dictators, Despots and Tyrants. Lanham [MD]: Rowman & Littlefield, 2006

Hattusilli the Third, king of the Hittites, tells in his autobiography that shortly after his brother became king there was a massive rebellion

Hattusilli then tells the story of one battle in which he claims he found the rebel general in the battlefield and personally killed him, which caused his army to run away and the rebellion to be defeated

Now, since this is an autobiography many could consider that Hattusilli is lying or exaggerating, but I think he was telling the truth, because if he wanted to exaggerate he could have added a lot more detail to make himself look better, but he basically just says: "...then I killed him and his army ran away. Anyway, then we were invaded by Ramses the Second..."

He is so casual about it I have no reason to believe it's propaganda

Before the Battle of Pratapgarh, Chhatrapati(essentially a title meaning emperor, but has a stronger connotation) Shivaji of the Maratha Empire attempted to negotiate with Afzal Khan, who was a General of the bijapur sultanate to avoid an imminent battle. Khan had no intentions of negotiating, and had cancelled a dagger on his person and intended to kill Chhatrapati Shivaji. Chhatrapati Shivaji knew not to trust him, and wore armor under his clothes and concealed a wagh nakha, which is a type of handheld weapon similar to brass knuckles but with sharp spikes attached to it.

When they met, Khan attempted to stab Chhatrapati Shivaji with the dagger, but was stopped by the armor, at which point Chhatrapati Shivaji used the wage nakha to disembowl and kill Khan. His troops then overran Khan's and he won the battle and liberated the land.

It was a Chhatrapati against a General, so not exactly what you described, but both were on a battlefield.