I was thinking that it has to do with accepting states in Touring Machine. You should just switch them so the complexity should be the same but I'm not sure.
That argument does not always work. You are probably thinking of the proofs that REG, R and RE are closed against complement.
Note how the same technique alread breaks down for CFL; it's not that easy for NPDA (or, in fact, for any non-deterministic automaton model).
This leads me to this answer: the flip-final-state technique works for membership in deterministic classes; I'm not sure if completeness always works out as well, but my gut feeling is yes.
The technique is unlikely to work even for membership in nondeterministic classes. As a matter of fact, we do not know if NP = co-NP and, as a consequence, we do not know if NPC = co-NPC.