I already told sassyfangs13 about this but I had a random Lutece AU pop into my head the other night.
Basically Robert is trying to convince Rosalind to fix the Anna Dewitt situation with few results. In essence Rosalind doesn’t trust Booker to be a good father (given he sold his daughter to wizard physicists ya know). Robert still issues his ultimatum so, as a compromise, Rosalind asks for time to evaluate Booker herself by going to a parallel universe to live a complete life with him. She returns and agrees to begin the experiment.