  We show that the non-embedded eigenvalues of the Dirac operator on the real
line with non-Hermitian potential $V$ lie in the disjoint union of two disks in
the right and left half plane, respectively, provided that the $L^1-norm$ of
$V$ is bounded from above by the speed of light times the reduced Planck
constant. An analogous result for the Schr\"odinger operator, originally proved
by Abramov, Aslanyan and Davies, emerges in the nonrelativistic limit. For
massless Dirac operators, the condition on $V$ implies the absence of nonreal
eigenvalues. Our results are further generalized to potentials with slower
decay at infinity. As an application, we determine bounds on resonances and
embedded eigenvalues of Dirac operators with Hermitian dilation-analytic
potentials.
