  We characterize manifolds which are locally conformally equivalent to either
complex projective space or to its negative curvature dual in terms of their
Weyl curvature tensor. As a byproduct of this investigation, we classify the
conformally complex space forms if the dimension is at least 8. We also study
when the Jacobi operator associated to the Weyl conformal curvature tensor of a
Riemannian manifold has constant eigenvalues on the bundle of unit tangent
vectors and classify such manifolds which are not conformally flat in
dimensions congruent to 2 mod 4.
