  The Aubry-Andr\'e-Harper (AAH) model with a self-dual symmetry plays an
important role in studying the Anderson localization. Here we find a self-dual
symmetry determining the quantum phase transition between extended and
localized states in a non-Hermitian AAH model and show that the eigenenergies
of these states are characterized by two types of winding numbers. By
constructing and studying a non-Hermitian generalized AAH model, we further
generalize the notion of the mobility edge, which separates the localized and
extended states in the energy spectrum of disordered systems, to the
non-Hermitian case and find that the generalized mobility edge is of a
topological nature even in the open boundary geometry in the sense that the
energies of localized and extended states exhibit distinct topological
structures in the complex energy plane. Finally, we propose an experimental
scheme to realize these models with electric circuits.
