In this paper, we study the complex
contact structure of a twistor space over a self-dual, Einstein
4-manifold with nonzero scalar curvature.
Although the existence of such a structure has been known
and well utilized by researchers
for several decades now, the Hermitian geometry resulting
from the complex contact structure is still in the
process of being fully developed. Here we
give a characterization
of such twistor spaces as those satisfying a curvature (and
hence purely geometric) identity. Later, we
describe how this result fits in with other areas
of research in complex contact geometry.