We extend the standard intertwining relations used in supersymmetrical (SUS
Y) quantum mechanics which involve real superpotentials to complex superpot
entials. This allows us to deal with a large class of non-Hermitian Hamilto
nians and to study in general the isospectrality between complex potentials
. In very specific cases we can construct in a natural way "quasscomppex" p
otentials which we define as complex potentials having a global property so
as to lead to a Hamiltonian with real spectrum. We also obtained a class o
f complex transparent potentials whose Hamiltonian can be intertwined to a
free Hamiltonian. We provide a variety of examples both for the radial prob
lem (half axis) and for the standard one-dimensional problem (the whole axi
s), including remarks concerning scattering problems.