Classification of the complement of a language $L = \{   \langle M \rangle \mid M \text{ accepts two strings of different lengths} \}$.

What is complement of this language?

my attempt:

the complement is: $M$ accepts no two strings of same length and this can be done by dovetailing different inputs on M but we can never say IT CANNOT ACCEPT TWO STRING OF SAME LENGTH and hence its NOT RECURSIVELY ENUMERABLE?
The complement of $L$ is:

$\bar{L} = \{\langle M \rangle \mid \text{ all strings accepted by $M$ have the same length} \}$

Your intuition that $\bar{L}$ is not recursively enumerable is correct, it is in fact complete for $\Pi^0_1$ by reduction from the emptiness problem. If you only need to prove that $\bar{L}$ is not in $\textsf{RE}$, it is sufficient to observe that $L$ is undecidable (reduction from the halting problem) but $\textsf{RE}$ (trivial).