Joking aside, what did you all take away from the story? I think Ill Saint made a lot of valid points except despite all those problems, for me the story did come together in the end. However due to the nature of the ending the discussion now seems to be largely about the specifics of time travel and multiverses. Once we can get past grasping the developer's idea of how all this works in their fictional world (and I think the OP has a reasonable explanation) isn't the next question about the themes and characters in the story? I'm still not sure what to make of "I'm both" and idea of choice/free will (and how it compares to the first Bioshock).
Even after all of the alternate timeless and parallel realities, I think the thing that interested me most about the story was the way the characters were gradually revealed to be different from how they first appeared, or in some cases, the way they changed before our eyes.
I particularly liked the way Booker and Elizabeth switched roles towards the end. It's so strange to me to think back on how I viewed Booker at the beginning of my first run; he seemed like a strong character, someone with some control and mastery over his life. He'd accepted a dangerous mission to a strange land, and he took it all in his stride. Elizabeth on the other hand began as a childlike figure, full of wonder and excitement at the world.
Cut to the end of the game and it's a different picture. Elizabeth has grown at a rate of knots, recreated as an almost world-weary woman with power and wisdom beyond any other force in the story. She takes control of not only her own life, but the lives of Booker and every other citizen of Columbia.
Booker on the other hand is shown for what he really is. A man who's committed atrocities for employers his whole life, eventually left broken and weary, descending into drink and gambling. By the time he reaches Columbia he's even more of a powerless chump than he was before, operating under the instructions of people he can't even remember meeting. In a story that focusses primarily on a girl kept locked away for her entire life, it's actually Booker who turns out the more pitiable. I found that fascinating.