SUSY quantum mechanics with complex superpotentials and real energy spectra

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.