  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.
