  In this paper, we extend the study of generalized Killing spinors on
Riemannian Spin$^c$ manifolds started by Moroianu and Herzlich to complex
Killing functions. We prove that such spinor fields are always real Spin$^c$
Killing spinors or imaginary generalized Spin$^c$ Killing spinors, providing
that the dimension of the manifold is greater or equal to 4. Moreover, we
classify Riemannian Spin$^c$ manifolds carrying imaginary and imaginary
generalized Killing spinors.
