  Topological phases have recently witnessed a rapid progress in non-Hermitian
systems. Here we study a one-dimensional non-Hermitian Aubry-Andr\'e-Harper
model with imaginary periodic or quasiperiodic modulations. We demonstrate that
the non-Hermitian off-diagonal AAH models can host zero-energy modes at the
edges. In contrast to the Hermitian case, the zero-energy mode can be localized
only at one edge. Such a topological phase corresponds to the existence of a
quarter winding number defined by eigenenergy in momentum space. We further
find the coexistence of a zero-energy mode located only at one edge and
topological nonzero energy edge modes characterized by a generalized Bott
index. In the incommensurate case, a topological non-Hermitian quasicrystal is
predicted where all bulk states and two topological edge states are localized
at one edge. Such topological edge modes are protected by the generalized Bott
index. Finally, we propose an experimental scheme to realize these
non-Hermitian models in electric circuits. Our findings add a new direction for
exploring topological properties in Aubry-Andr\'e-Harper models.
