  Let M be a pseudo-Riemannian manifold with a pseudo-Hermitian complex
structure $J$. We give necessary and sufficient conditions that the curvature
operator $R(\pi)$ is complex linear when $\pi$ is a $J$ invariant real 2 plane.
Under this assumption, we study when M is complex IP - i.e. the spectrum, or
more generally the Jordan normal form, of $R(\pi)$ is constant on the
Grassmannian of complex spacelike or timelike lines. Methods from algebraic
topology are used to obtain restrictions on the spectrum of a complex IP
algebraic curvature tensor.
