If $A$ is m-complete for $\Sigma_n$, is $\overline{A}$ m-complete for $\Pi_n$?
Hint: use the fact that $A\le_p B \iff \overline{A} \le_p \overline{B}$ (the same reduction works for both sides). 

Try proving this for $NP,coNP$ first, and then check if your proof generalizes to the n'th level too.