  We wish to attack the problems that H.~Anciaux and K.~Panagiotidou posed in
[1], for non-degenerate real hypersurfaces in indefinite complex projective
space. We will slightly change these authors' point of view, obtaining cleaner
equations for the almost contact metric structure. To make the theory
meaningful, we construct new families of non-degenerate Hopf real hypersurfaces
whose shape operator is diagonalisable, and one Hopf example with degenerate
metric and non-diagonalisable shape operator. Next, we obtain a rigidity
result. We classify those real hypersurfaces which are $\eta$-umbilical. As a
consequence, we characterize some of our new examples as those whose Reeb
vector field $\xi$ is Killing.
