  We show that allowing magnetic fields to be complex-valued leads to an
improvement in the magnetic Hardy-type inequality due to Laptev and Weidl. The
proof is based on the study of momenta on the circle with complex magnetic
fields, which is of independent interest in the context of PT-symmetric and
quasi-Hermitian quantum mechanics. We study basis properties of the
non-self-adjoint momenta and derive closed formulae for the similarity
transforms relating them to self-adjoint operators.
