  In principle, non-Hermitian quantum equations of motion can be formulated
using as a starting point either the Heisenberg's or the Schr\"odinger's
picture of quantum dynamics. Here it is shown in both cases how to map the
algebra of commutators, defining the time evolution in terms of a non-Hermitian
Hamiltonian, onto a non-Hamiltonian algebra with a Hermitian Hamiltonian. The
logic behind such a derivation is reversible, so that any Hermitian Hamiltonian
can be used in the formulation of non-Hermitian dynamics through a suitable
algebra of generalized (non-Hamiltonian) commutators. These results provide a
general structure (a template) for non-Hermitian equations of motion to be used
in the computer simulation of open quantum systems dynamics.
