How to switch from a physics PhD. to complexity theory?
I would suggest you to continue studying in your physics Ph.D. program and successfully graduate. However, considering your interest in complexity theory (which I share with you, by the way, though, unfortunately, my math background is more limited than yours), I would advise to find a topic on the intersection of physics and complexity theory (aka complex systems) and write dissertation on that topic. That way will have a nice and flexible general physics education and, at the same time, you will stay motivated (which is extremely important for writing a dissertation) throughout the whole dissertation writing period. As for the complex systems field of study, please see my relevant question on Cross Validated site and references within. Good luck to both of us!