  Motivated by what one observes dealing with PT-symmetric quantum mechanics,
we discuss what happens if a physical system is driven by a diagonalizable
Hamiltonian with not all real eigenvalues. In particular, we consider the
functional structure related to systems living in finite-dimensional Hilbert
spaces, and we show that certain intertwining relations can be deduced also in
this case if we introduce suitable antilinear operators. We also analyze a
simple model, computing the transition probabilities in the broken and in the
unbroken regime.
