Here's how I understood it:
Booker is born, grows up, and takes part in the massacre at Wounded Knee while he's in the Army.
Booker has daughter, Anna, but when asked where she came from he can't come up with an explanation.
Booker takes up drinking and gambling to ease his guilt over what happened at Wounded Knee, and amasses a substantial debt.
Booker is approached by a representative of a wealthy industrialist who offers to buy his daughter in exchange or wiping out Booker's debt. Booker agrees, but at the last moment tries to go back on the deal. He's unable to get his daughter back and carves her initials into his hand as a form of penance.
Booker's guilt and depression become worse and he turns to religion. He's baptized and takes a new name, Comstock.
Comstock decides to live in Columbia, where he becomes a major political/religious figurehead.
While living in Columbia, Comstock works with another group of people on a time machine/gateway to parallel universes. An unforeseen side effect of this work renders Comstock sterile, but the project is a success and the machine works.
Still feeling guilt about his decision to sell his daughter and now unable to have any more children, Comstock travels back in time/space and buys Booker's daughter. He brings her to Columbia and gives her a new name: Elizabeth.
The Booker from the past who just sold his daughter goes through the same religious conversion but rejects baptism at the last moment. He's later approached by the Luteces (I missed why), who bring him into the dimension where Elizabeth/Anna now lives. The process messes with Booker's memories and he fabricates the "Save the girl, wipe out the debt" scenario from what little pieces of information he managed to retain.
Booker goes through the events of the game with Elizabeth, which results in the destruction of Columbia. Elizabeth's/Anna's accident (her finger getting cut off as the portal closes) back when Booker originally sold her is what gave her the abilities to see into other worlds/timelines (the tears). With the the device that was limiting her powers is destroyed as well, she's able to see that by Booker purchasing her from himself, he's created an endless series of time loops/universes where the events will play out again and again and again.
The Elizabeths/Annas bring Booker to the exact moment where he is no longer the man he was before the baptism, but is not yet one of the two men he will eventually become. They drown him there, sealing off the loops and preventing the whole mess (and themselves) from existing in the first place.
What I don't understand is the post credit scene. Has the loop not been broken, or is a new path/chance available, or what?
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