  The aim of this paper is to prove a complex version of the main theorem in
\cite{MY}. We will prove that among pairs $K, K' \subset \mathbb{C}$ of
conformal dynamically defined Cantor sets with sum of Hausdorff dimension
$HD(K)+HD(K') > 2$, there is an open and dense subset of such pairs verifying
$\text{int}(K-K') \neq \emptyset$.
