Redditor u/grambocrackah posits an intriguing theory that offers an explanation for why Alexei didn't merely break out of prison. With the super-soldier serum, he's certainly strong enough to overpower the guards and likely break out whenever the mood fancies him. The theory also states that the prison probably isn't too far away from a town, seeing how Alexei, Natasha, and Yelena make it to Melina's farm on just a quarter tank of fuel. So why would he essentially choose to stay in prison?
The user goes on to mention, "I believe that whatever process or formula turned Alexei into the Red Guardian is also gradually killing him, and he needs constant medical intervention to stay alive. He gets that in prison, and he's either too uninformed or under-equipped to handle it by himself, which is why he stays. It's also the reason he's gone out of his way to know exactly where Melina is — she managed his treatment in Ohio and is one of the few people on the outside that may be able and willing to help him."
Granted, there's not much evidence to support this theory in the movie, but the user does point out how we might already see the effects of his deteriorating health. Alexei hasn't stayed in shape, as Melina points out when they reunite. He also has bad skin, terrible teeth, and a seemingly declined mental state. All this could be chalked up to the serum taking its toll on his body and mind. It would also explain why he wants to see Melina after the breakout; he knows she's one of the only people that can help him now that he's out of prison. While it's just a theory, it could add some interesting wrinkles to Alexei's story if he pops up again in the Marvel Cinematic Universe.
