Prove that there exists an undecidable complementary language
Take $\Sigma=\{0,1\}$ for simplicity.

Define $A \oplus B = 0A + 1B$. We have that $(A \oplus B)^c \equiv_m A^c \oplus B^c$ since they differ only by the empty string.

We now take any non recursive $A$, and consider $B = A \oplus A^c$. We have $A \leq_m B$, so $B$ is not recursive as well. Further, $B = A \oplus A^c \equiv_m (A^c \oplus A)^c \equiv_m (A \oplus A^c)^c = B^c$.