  In this paper the geometric theory of separation of variables for
time-independent Hamilton-Jacobi equation is extended to include the case of
complex eigenvalues of a Killing tensor on pseudo-Riemannian manifolds. This
task is performed without to complexify the manifold but just considering
complex-valued functions on it. The simple formalism introduced allows to
extend in a very natural way the classical results on separation of variables
(including Levi-Civita criterion and Stackel-Eisenhart theory) to the complex
case. Orthogonal variables only are considered.
