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After the events at Wounded Knee, Booker went to the baptism looking to cleanse himself of his sins. In one universe he accepted the baptism and became Comstock.

In another universe, he rejected the baptism and continued as the Booker we end up playing. In the universes where he made this choice, he eventually had a wife and a daughter. He had spent his life drinking and gambling to try and forget the events of Wounded Knee as well as the death of his wife. As a result he ended up with a large amount of gambling debt.

He was contacted by Robert Lutece who offered to wipe away the debt in exchange for his daughter. This is actually the origin of the "bring us the girl and wipe away the debt" phrase. Booker sold his daughter to Lutece and Comstock, but had a change of heart and tried to get her back before Robert and Comstock took her through the tear into their universe. This is how Anna/Elizabeth's pinky gets severed and is why she has the ability to open tears.

So why did Comstock want Anna? Well, this came about because of the prophecy you hear in the game: "the seed of the prophet shall sit the throne and down in flame the mountains of man." Basically that prophecy says that Comstock's child will be the one to carry out his legacy of "destroying the Sodom below."

There was one big problem, though: Comstock was made sterile by prolonged exposure to the tear technology that Rosalind Lutece had developed. Rosalind had actually contacted her alternate reality self (Robert) and wanted to bring him to her universe. Simultaneously Comstock needed a child and Rosalind came up with a plan to take the child of an alternate reality Comstock/Booker. Technically that would be Comstock's child.

Eventually Robert Lutece began to feel bad that they had abducted Anna from Booker and set the events of the game into motion. They bring Booker into a Comstock universe so that he can get his daughter back and try to end the whole thing with Comstock. There are hints all throughout the game that Booker is not from the universe with Columbia, because he knows nothing about Columbia.