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In this paper, a one-dimensional non-Hermitian quasiperiodic $p$-wavesuperconductor without $\mathcal{PT}$-symmetry is studied. By analyzing thespectrum, we discovered there still exists real-complex energy transition evenif the inexistence of $\mathcal{PT}$-symmetry breaking. By the inverseparticipation ratio, we constructed such a correspondence that pure realenergies correspond to the extended states and complex energies correspond tothe localized states, and this correspondence is precise and effective todetect the mobility edges. After investigating the topological properties, wearrive at a fact that the Majorana zero modes in this system are immune to thenon-Hermiticity.

Majorana zero modes, unconventional real-complex transition and mobility edges in a one-dimensional non-Hermitian quasi-periodic lattice