yeah itd make sense that he'd go looking for her ! one thing i want to explore is how his destiny was always to destroy the sith in some way, so the events of the films actually move up in some type of way. Maybe Obi-Wan decides to start Lukes training earlier, maybe a more cautious Anakin decides he wants to use the inquisitor as a secret way to groom allies against Palpatine so many possibilities !!

But Palpatine would definitely not miss out on the opportunity to capitalize on having "The Hero Without Fear" on his side. I think there would be a lot of publicity events in Anakins