I used to wonder the same thing myself. The behavior you describe seems relatively common in recently stocked trout, I've seen it many times. As best as I can figure, they are fighting for dominance and/or a particular spot in the stream. I can remember seeing trout do this in the Mill River in Hamden, CT, shortly after they were stocked. Sometimes they appear to be playing, but I'm pretty sure it's not the case. Usually the two fish are close to the same size, and I think this is how they determine who the boss is, the "alpha" trout. If one was significantly larger, the smaller one probably wouldn't challenge the other.
Once I caught a 20" golden trout outside of a catch & release area I used to frequent- it had moved downstream over a low dam during high water. I knew that if I put it back right where I caught it, that come Opening Day it would be gone within a week- it hung out right where you could see it. So next time I came back I caught it again, put it in a big bucket, and transported it back upstream into the catch & release area. There was another golden (they were both males) about the same size in the pool, and the two trout immediately started fighting- chasing each other, butting with their heads, and biting. Again, each trying to assert it's dominance in the pool, so as to be able to choose the best feeding lie. For the rest of the time I was there, all they did was fight non-stop, and neither would even look at any fly I threw at them. Maybe a giant golden trout streamer would have worked?