  We investigate the localization and topological transitions in a
one-dimensional (interacting) non-Hermitian quasiperiodic lattice, which is
described by a generalized Aubry-Andr\'{e}-Harper model with irrational
modulations in the off-diagonal hopping and on-site potential and with
non-Hermiticities from the nonreciprocal hopping and complex potential phase.
For noninteracting cases, we reveal that the nonreciprocal hopping (the complex
potential phase) can enlarge the delocalization (localization) region in the
phase diagrams spanned by two quasiperiodical modulation strengths. We show
that the localization transition are always accompanied by a topological phase
transition characterized the winding numbers of eigenenergies in three
different non-Hermitian cases. Moreover, we find that a real-complex
eigenenergy transition in the energy spectrum coincides with (occurs before)
these two phase transitions in the nonreciprocal (complex potential) case,
while the real-complex transition is absent under the coexistence of the two
non-Hermiticities. For interacting spinless fermions, we demonstrate that the
extended phase and the many-body localized phase can be identified by the
entanglement entropy of eigenstates and the level statistics of complex
eigenenergies. By making the critical scaling analysis, we further show that
the many-body localization transition coincides with the real-complex
transition and occurs before the topological transition in the nonreciprocal
case, which are absent in the complex phase case.
