This article assumes you've played the entirety if Bioshock Infinite (not counting the DLC as I've never played that). Spoilers ahead.
There are a number of things that don't make much sense in Infinite. There are timeline errors that are particularly confusing. I won't bore you with the details. You either are aware of it, or you can Google it and find more detailed explanations than I can provide.
But really, this game is all over the map. It has a ghost mom who is raised from the dead, for Pete's sake.
I've spent a great deal of time in the last several years pondering this game. Much more than I should have, probably. I was thinking about this game in the shower this morning when I had an epiphany. I'm sure I'm not the first person to come up with this theory, but I've never encountered it before. So it was all new to me. But here's the idea: The game takes place entirely in the mind of Booker DeWitt.
Booker is a broken man. He was in the military in the Boxer Rebellion and he was at Wounded Knee. He was also in the Pinkerton agency. He has seen and done some serious shit. This has obviously had a huge effect upon him. He likely has some form of PTSD. We know he has a severe gambling problem. He also appears to be an alcoholic. It's even possible that he was involved with other drugs in some form, but we don't really know. The point is, he's messed up. His brain is addled by trauma and booze. He has guilt and fear. He's reached a breaking point. His gambling debts have come back to haunt him. He is afraid he'll lose everything. This probably wouldn't bother him so much if it weren't for one element of his life: Anna.
Booker has a daughter. Anna is an infant that he is responsible for. He is in charge of loving, raising, protecting, and providing for this baby. But his life is disintegrating around him. He is unqualified and incapable of providing for this girl. One night, in a drunken frenzy, he snaps.
Thus we enter the world of Bioshock Infinite. Booker can't handle the reality in which he lives. The pain, guilt, and fear have overpowered him. He looks back on his life and thinks if what might have been. As he looks back, he remembers one specific moment from his past that could have changed his whole life. After Wounded Knee, he encountered a minister. This minister was baptizing people after the battle. In reality, Booker thought about this and ultimately rejected the opportunity. He felt that nothing could wipe away what he had done. But in this fantasy, he imagines what may have happened if he had been baptized. What kind of man might he have become.
He envisions himself as a visionary man. A great leader. A religious Titan. A man so inspiring that he literally lifts an entire city into the air.
But then things start to cloud the fantasy.
Obviously Booker is intellectually familiar with Christianity. He even knows some folk hymns. He is aware of biblical symbols. He is also knowledgeable about American history. His mind warps these two concepts together into this quasi religion that he is the master of. But his own weaknesses and insecurities darken it. The fantasy introduces institutionalized racism. Economic disparity. Slavery. And of course, the one thing that has defined his whole life: violence.
Even in his fantasy life, he realizes that he would not be a perfect man. Even as a religious leader, he would fail. He would create a system that does more harm than good. Even changing his name and his appearance can't undo the fact that Booker will do nothing but hurt the people around him.
Beyond all of this, there is one fundamental flaw in this fantasy that is more personally problematic for Booker. If he had been baptized and completely changed his life, he wouldn't have Anna. His daughter would not exist. The baby girl who is, in reality, only one room away, would be gone.
This is what the story is about. Booker has a daughter. But he can't have a daughter. He is a drunk. He is emotionally damaged. He is a gambler. He is broke. He contributes nothing of value to the world and he is suddenly responsible for the most precious and valuable thing in the world: a baby.
At this point it is worth mentioning Anna's mother. As far as I can recall, the game doesn't really address who the mother was and why Booker must raise Anna without her. Did she abandon her child? Did she die? I'm inclined to think the latter.
In the fantasy of the game, when Booker becomes Comstock, he has a wife. He has found love. But in this fantasy, his beautiful and amazing wife can't produce the one meaningful thing in his reality: Anna.
In the fantasy, the wife dies. I personally think this reflects that Anna's mother died in reality. That's why Booker had the mental breakdown. Before the mother's death, he knew that she would provide for Anna. He could love the child from afar and not bring pain into her life because her mother would protect her. But then the mother died, forcing Anna into the hands of Booker.
As the game progresses, we experience a battle between the reality of Booker and the fantasy of Comstock. In the end, Booker gives up the fantasy. He actually kills Comstock. But he goes further. He goes back in his mind to the place where it all started: the baptism at Wounded Knee. He erases this from his mind. He finally accepts that he can't hinge his life on this decision. He must take his life as it is. He can't blame the past for his problems, and he can't look to the past for relief.
Then, after the credits, Booker is home. He hears music coming from Anna's room and he rushes to her, calling out her name. Booker has finally accepted that his reality is the life he must live. And he must live it for Anna.