The usual candidates for context free languages whose complement is also context free, but they are not regular are the Deterministic Context Free Languages ($DCFL$).
For example, $L=\{a^nb^n\mid n\in\mathbb N\}$ is such language.
Is there a language, $L\in CFL\setminus DCFL$ such that $\bar L\in CFL$?
Since $DCFL$ is closed under complement, this is equivalent of asking whether $$CFL\cap co\text{-}CFL = DCFL$$