  We investigate the classical limit of non-Hermitian quantum dynamics arising
from a coherent state approximation, and show that the resulting classical
phase space dynamics can be described by generalised "canonical" equations of
motion, for both conservative and dissipative motion. The dynamical equations
combine a symplectic flow associated with the Hermitian part of the Hamiltonian
with a metric gradient flow associated with the anti-Hermitian part of the
Hamiltonian. We derive this structure of the classical limit of quantum systems
in the case of a Euclidean phase space geometry. As an example we show that the
classical dynamics of a damped and driven oscillator can be linked to a
non-Hermitian quantum system, and investigate the quantum classical
correspondence. Furthermore, we present an example of an angular momentum
system whose classical phase space is spherical and show that the generalised
canonical structure persists for this nontrivial phase space geometry.
