The non-Hermitian quadratic oscillator studied by Swanson is one of the
popular $PT$-symmetric model systems. Here a full classical description of its
dynamics is derived using recently developed metriplectic flow equations, which
combine the classical symplectic flow for Hermitian systems with a dissipative
metric flow for the anti-Hermitian part. Closed form expressions for the metric
and phase-space trajectories are presented which are found to be periodic in
time. Since the Hamiltonian is only quadratic the classical dynamics exactly
describes the quantum dynamics of Gaussian wave packets. It is shown that the
classical metric and trajectories as well as the quantum wave functions can
diverge in finite time even though the $PT$-symmetry is unbroken, i.e., the
eigenvalues are purely real.