  We provide a new perspective on non-Hermitian evolution in quantum mechanics
by emphasizing the same method as in the Hermitian quantum evolution. We first
give a precise description of the non unitary evolution, and collecting the
basic results around it and postulating the norm preserving. This cautionary
postulate imposing that the time evolution of a non Hermitian quantum system
preserves the inner products between the associated states must not be read
naively. We also give an example showing that the solutions of time-dependent
non Hermitian Hamiltonian systems given by a linear combination of SU(1,1) and
SU(2) are obtained thanks to time-dependent non-unitary transformation.
