How to prove that the Myhill-Nerode equivalence classes for L are the same as for its complement?
Note that the Myhill-Nerode equivalences are also defined for non-regular languages, so constructing a finite state automaton will not help.

What is the definition of such a class? $x \sim_L y$ iff $xz\in L$ whenever $yz\in L$ (for any string $z$). How does that change when we consider the complement of $L$? We use basic logic: $P\Leftrightarrow Q $ iff $\lnot P\Leftrightarrow \lnot Q $.

$x \sim_L y$
iff  $xz\in L$ whenever $yz\in L$ (for any string $z$)
iff  (not $xz\in L$) whenever (not $yz\in L$) (for any string $z$)
iff  $xz\in \bar L$ whenever $yz\in \bar L$ (for any string $z$)
iff  $x \sim_{\bar L} y$.