  We theoretically investigate topological features of a one-dimensional
Su-Schrieffer-Heeger lattice with modulating non-Hermitian on-site potentials
containing four sublattices per unit cell. The lattice can be either
commensurate or incommensurate. In the former case, the entire lattice can be
mapped by supercells completely. While in the latter case, there are two extra
lattice points, thereby making the last cell incomplete. We find that an
anti-PT transition occurs at exceptional points of edge states at certain
parameters, which does not coincide with the conventional topological phase
transition characterized by the Berry phase, provided the imaginary on-site
potential is large enough. Interestingly, when the potential exceeds a critical
value, edge states appear even in the regime with a trivial Berry phase. To
characterize these novel edge states we present topological invariants
associated with the system's parity. Finally, we analyze the dynamics for
initial states with different spatial distributions, which exhibit distinct
dynamics for the commensurate and incommensurate cases, depending on the
imaginary part of edge state energy.
