  The complex geometry underlying the Schr\"odinger dynamics of coherent states
for non-Hermitian Hamiltonians is investigated. In particular two seemingly
contradictory approaches are compared: (i) a complex WKB formalism, for which
the centres of coherent states naturally evolve along complex trajectories,
which leads to a class of complexified coherent states; (ii) the investigation
of the dynamical equations for the real expectation values of position and
momentum, for which an Ehrenfest theorem has been derived in a previous paper,
yielding real but non-Hamiltonian classical dynamics on phase space for the
real centres of coherent states. Both approaches become exact for quadratic
Hamiltonians. The apparent contradiction is resolved building on an observation
by Huber, Heller and Littlejohn, that complexified coherent states are
equivalent if their centres lie on a specific complex Lagrangian manifold. A
rich underlying complex symplectic geometry is unravelled. In particular a
natural complex structure is identified that defines a projection from complex
to real phase space, mapping complexified coherent states to their real
equivalents.
