  The chiral anomaly underlies a broad number of phenomena, from enhanced
electronic transport in topological metals to anomalous currents in the
quark-gluon plasma. The discovery of topological states of matter in
non-Hermitian systems -- effective descriptions of dissipative systems --
raises the question of whether there are anomalous conservation laws that
remain unaccounted for. To answer this question, we consider both $1+1$ and
$3+1$ dimensions, presenting a unified formulation to calculate anomalous
responses in Hermitianized, anti-Hermitianized and non-Hermitian systems of
massless electrons with complex Fermi velocities coupled to non-Hermitian gauge
fields. Our results indicate that the quantum conservation laws of chiral
currents of non-Hermitian systems are not related to those in Hermitianized and
anti-Hermitianized systems, as would be expected classically, due to novel
anomalous terms that we derive. These may have implications for a broad class
of emerging experimental systems that realize non-Hermitian Hamiltonians.
