complexity theory – Is there a non-deterministic polynomial by time Turing machine such that: $L(M)in NPC$ and $L(overline{M})in P$

When $overline{M}$ is a non-deterministic polynomial by time Turing machine that final states switched: accept to reject and vice versa.
I’m thinking that this equal to $P=NP$, but I saw a solution (an example) that I disagree with:
$M$ is a non-deterministic polynomial by time Turing machine that decide $SAT$, if all that paths are rejected then $L(overline{M})=Sigma^*in P$

Is it a valid solution, or as I’m thinking $L(M)in NPC$ and $L(overline{M})in P Leftrightarrow P=NP$