  Non-Hermitian topological phases have gained immense attention due to their
potential to unlock novel features beyond Hermitian bounds. PT-symmetric
(Parity Time-reversal symmetric) non-Hermitian models have been studied
extensively over the past decade. In recent years, the topological properties
of general non-Hermitian models, regardless of the balance between gains and
losses, have also attracted vast attention. Here we propose a non-Hermitian
second-order topological (SOT) insulator that hosts gapless corner states on a
two-dimensional quasi-crystalline lattice (QL). We first construct a
non-Hermitian extension of the Bernevig-Hughes-Zhang (BHZ) model on a QL
generated by the Amman-Beenker (AB) tiling. This model has real spectra and
supports helical edge states. Corner states emerge by adding a proper Wilson
mass term that gaps out the edge states. We propose two variations of the mass
term that result in fascinating characteristics. In the first variation, we
obtain a purely real spectra for the second-order topological phase. In the
latter, we get a complex spectra with corner states localized at only two
corners. Our findings pave a path to engineering exotic SOT phases where corner
states can be localized at designated corners.
