We analyze some features of alternative Hermitian and quasi-Hermitian quantum descriptions of simple and bipartite compound systems. We show that alternative descriptions of two interacting subsystems are possible if and only if the metric operator o
f the compound system can be obtained as tensor product of positive operators on component spaces. Some examples also show that such property could be strictly connected with symmetry properties of the non-Hermitian Hamiltonian.
We show that the complex projections of time-dependent $eta $-quasianti-Hermitian quaternionic Hamiltonian dynamics are complex stochastic dynamics in the space of complex quasi-Hermitian density matrices if and only if a quasistationarity condition
is fulfilled, i. e., if and only if $eta $ is an Hermitian positive time-independent complex operator. An example is also discussed.
