  We investigate a non-Hermitian Aubry-Andr\'e-Harper model with short-range,
as well as long-range p-wave pairing. Here, the non-Hermiticity is introduced
through the onsite potential. A comprehensive analysis of several critical
aspects of this system is conducted, which includes eigenspectra, topological
properties, localization properties, and the transition from real to complex
energies. Specifically, we observe the emergence of Majorana zero modes in the
case of short-range pairing, whereas massive Dirac modes emerge in the case of
long-range pairing. More importantly, for the case of short-range pairing, we
observe two simultaneous phase transitions or double phase transitions:
topological and multifractal to localized phase. On the other hand, in the case
of the long-range pairing, the topological and multifractal to localized
transitions do not coincide. However, for both ranges of pairing, we identify a
double phase transition where delocalized (or metallic) to a critical
multifractal state is accompanied by an unconventional shift from real to
complex energies. Unlike the short-range pairing case, we observe mobility
edges in the long-range pairing case.
