  Interpolating between the classic notions of intersection and polar centroid
bodies, (real) $L_p$-intersection bodies, for $-1<p<1$, play an important role
in the dual $L_p$-Brunn--Minkowski theory. Inspired by the recent construction
of complex centroid bodies, a complex version of $L_p$-intersection bodies,
with range extended to $p>-2$, is introduced, interpolating between complex
intersection and polar complex centroid bodies. It is shown that the complex
$L_p$-intersection body of an $\mathbb{S}^1$-invariant convex body is
pseudo-convex, if $-2<p<-1$ and convex, if $p\geq-1$. Moreover, intersection
inequalities of Busemann--Petty type in the sense of
Adamczak--Paouris--Pivovarov--Simanjuntak are deduced.
