  We study the geometry of null hypersurfaces in indefinite complex contact
manifolds. We prove several classification results for a variety of well-known
null hypersurfaces, including the totally umbilic, totally screen umbilic, and
the screen conformal ones. Furthermore, a characterization of the ambient space
is given in case the underlying null hypersurface is totally contact umbilic,
totally contact screen umbilic or contact screen conformal, i.e. we have proved
that the ambient complex contact manifold must be a space of constant
$GH$-sectional curvature of $-3$.
