  Both non-Hermitian systems and the behaviour of emitters coupled to
structured baths have been studied intensely in recent years. Here we study the
interplay of these paradigmatic settings. In a series of examples, we show that
a single quantum emitter coupled to a non-Hermitian bath displays a number of
unconventional behaviours, many without Hermitian counterpart. We first
consider a unidirectional hopping lattice whose complex dispersion forms a
loop. We identify peculiar bound states inside the loop as a manifestation of
the non-Hermitian skin effect. In the same setting, emitted photons may display
spatial amplification markedly distinct from free propagation, which can be
understood with the help of the generalized Brillouin zone. We then consider a
nearest-neighbor lattice with alternating loss. We find that the long-time
emitter decay always follows a power law, which is usually invisible for
Hermitian baths. Our work points toward a rich landscape of anomalous quantum
emitter dynamics induced by non-Hermitian baths.
