  This article is a contribution to the study of superintegrable Hamiltonian
systems with magnetic fields on the three-dimensional Euclidean space
$\mathbb{E}_3$ in quantum mechanics. In contrast to the growing interest in
complex electromagnetic fields in the mathematical community following the
experimental confirmation of its physical relevance [X. Peng et al., Phys. Rev.
Lett. 114 (2015)], they were so far not addressed in the growing literature on
superintegrability. Here we venture into this field by searching for additional
first order integrals of motion to the integrable systems of cylindrical type.
We find that already known systems can be extended into this realm by admitting
complex coupling constants. In addition to them, we find one new system whose
integrals of motion also feature complex constants. All these systems are
multiseparable. Rigorous mathematical analysis of these systems is challenging
due to the non-Hermitian setting and lost gauge invariance. We proceed formally
and pose the resolution of these problems as an open challenge.
