The answer, in true quantum confuddlery, is both can happen. Sometimes the Bookers and Elizabeths are the same, other times they are different and a duplicate exists. For instance, when they move to the universe where DeWitt became a Vox Populi hero but died, Booker's memory collapses once more. However, the earlier jump didn't affect his memory so much.|
However, this and the explanation of Elizabeht's powers are the weakest and most hand-wavey parts of the story. They will not hold up to scrutiny if you consider every aspect, no matter how you wash it. Just don't bother trying.
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