Complex IP curvature tensors

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.