Quasistationary quaternionic Hamiltonians and complex stochastic maps

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.