  We complete the classification of six-dimensional strongly unimodular almost
nilpotent Lie algebras admitting complex structures. For several cases we
describe the space of complex structures up to isomorphism. As a consequence we
determine the six-dimensional almost nilpotent solvmanifolds admitting an
invariant complex structure and study the existence of special types of
Hermitian metrics, including SKT, balanced, locally conformally K\"ahler, and
strongly Gauduchon metrics. In particular, we determine new balanced
solvmanifolds and confirm a conjecture by the first author and Vezzoni
regarding SKT and balanced structures in the six-dimensional strongly
unimodular almost nilpotent case. Moreover, we prove some negative results
regarding complex structures tamed by symplectic forms, showing in particular
that in every dimension such structures cannot exist on non-K\"ahler almost
abelian Lie algebras.
