  The correspondence between the real-complex transition in energy and
delocalization-localization transition is well-established in a class of
Aubry-Andr'e-Harper model with exponential non-Hermitian on-site potentials. In
this paper, we study a generalized Aubry-Andr'e model with off-diagonal
modulation and non-Hermitian on-site potential. We find that, when there exists
an incommensurate off-diagonal modulation, the correspondence breaks down,
although the extended phase is maintained in a wide parameter range of the
strengths of the on-site potential and the off-diagonal hoppings. An additional
intermediate phase with a non-Hermitian mobility edge emerges when the
off-diagonal hoppings become commensurate. This phase is characterized by the
real and complex sections of the energy spectrum corresponding to the extended
and localized states. In this case, the aforementioned correspondence reappears
due to the recovery of the PT-symmetry.
