  We demonstrate the existence of generalized Aubry-Andr\'e self-duality in a
class of non-Hermitian quasi-periodic lattices with complex potentials. From
the self-duality relations, the analytical expression of mobility edges is
derived. Compared to Hermitian systems, mobility edges in non-Hermitian ones
not only separate localized from extended states, but also indicate the
coexistence of complex and real eigenenergies, making it possible a topological
characterization of mobility edges. An experimental scheme, based on optical
pulse propagation in synthetic photonic mesh lattices, is suggested to
implement a non-Hermitian quasi-crystal displaying mobility edges.
