  The complex eigenvalues of some non-Hermitian Hamiltonians, e.g. parity-time
symmetric Hamiltonians, come in complex-conjugate pairs. We show that for
non-Hermitian scattering Hamiltonians (of a structureless particle in one
dimension) possesing one of four certain symmetries, the poles of the
$S$-matrix eigenvalues in the complex momentum plane are symmetric about the
imaginary axis, i.e. they are complex-conjugate pairs in complex-energy plane.
This applies even to states which are not bounded eigenstates of the system,
i.e. antibound or virtual states, resonances, and antiresonances. The four
Hamiltonian symmetries are formulated as the commutation of the Hamiltonian
with specific antilinear operators. Example potentials with such symmetries are
constructed and their pole structures and scattering properties are calculated.
