Firstly, the sentence you quoted, while unfortunately ambiguous, is correct if you take "of which" as picking up "assumptions" instead of "results". That is, the assumptions, of which a famous one is AC, have counterintuitive consequences, of which the original poster does not give an example but Banach–Tarski is a commonly cited one.

Secondly, the axiom of choice may be independent of ZF (depending on whether ZF is consistent) but it is certainly not independent of ZFC :).