I wrote a comment on pretty much the exact same question a few weeks back, but I'm on mobile and can't easily get to it--when I get the chance I'll track it down and post it (EDIT: What luck, I found it!). The short version is yes, naumachiae were very much a thing (Caesar held the first one during his triumph in 46). But amphitheaters were rarely flooded. Naumachiae generally took place in man-made basins--Caesar built a giant temporary basin in the Campus Martius that was filled when the games were over, and Augustus built a huge one near the Tiber island that was still in use when Titus staged games there as part of the dedication for the Flavian Amphitheater, also known as the Colosseum. The largest naumachiae ever were Claudius', staged in the Fucine Lake before he drained it. As far as amphitheaters go, they were too small to stage anything but the smallest naumachiae--these were spectacles that needed huge amounts of room (Augustus' naumachiae featured thirty triremes and biremes, as well as multiple small vessels and thousands of rowers and marines). Until the Colosseum was built permanent amphitheaters didn't really exist in the city, which made it all the more difficult to stage them in anything but a basin. The first naumachiae in an amphitheater that I know of are Nero's, which he held in a wooden amphitheater built on the Campus Martius--the amphitheater was drained afterward and regular combats held afterward, until the amphitheater was torn down at the end of thr games. Suetonius claims that it was filled with seawater, which is not in any of our other sources--I'm more than a bit skeptical of that, since apart from the difficulties of bringing so much seawater, the location of the theater on the Campus Martius suggests that, like Caesar, Nero just filled it by connecting it to the Tiber (and we know that it was connected somehow, because he drained it into the Tiber). Titus held small naumachiae in the Colosseum during its dedication ceremony, but the big ones were held at Augustus' basin, where he set up a false floor and held simultaneous land and sea battles. As I recall the only time that we know of when the Colosseum was used to stage naumachiae other than this is briefly under Domitian, during the dedication of his amphitheater

Did they ever put alligators or crocodiles into the water to make things more interesting?