Is there a non-deterministic polynomial by time Turing machine such that: $L(M)\in NPC$ and $L(\overline{M})\in P$
Let's imagine a Non Deterministic Turing Machine $M$ that decide $SAT$. If we tune this machine a bit, and add a transition on the initial state, for every letter read, that reject the entry. Let $M'$ be the new NTM. Then, $L(M') = L(M)$, since $u\in L(M) \Leftrightarrow \exists$ at least one computational path in $M$ to an accepting state (and the same thing for $M'$).
Now consider $\overline{M'}$. Since we added a rejecting transition which is possible for every entry in $M'$, that means that there is an accepting transition for every entry in $\overline{M'}$, so it means that $L(\overline{M'}) = \Sigma^* \in P$.
That does not necessarily imply that $P = NP$, the reason being that $L(\overline{M'}) \neq \overline{L(M')}$.
Generaly speaking, $L \in NPC \Leftrightarrow \overline{L} \in \text{co-}NPC$, but that is not the case if you consider a Turing Machine and its switched version.