  Anderson $\textit{et al}$ have shown that for complex energies, the classical
trajectories of $\textit{real}$ quartic potentials are closed and periodic only
on a discrete set of eigencurves. Moreover, recently it was revealed that, when
time is complex $t$ $(t=t_{r}e^{i\theta _{\tau }}),$ certain real hermitian
systems possess close periodic trajectories only for a discrete set of values
of $\theta _{\tau }$. On the other hand it is generally true that even for real
energies, classical trajectories of non $\mathcal{PT}$- symmetric Hamiltonians
with complex parameters are mostly non-periodic and open. In this paper we show
that for given real energy, the classical trajectories of $\textit{complex}$
quartic Hamiltonians $H=p^{2}+ax^{4}+bx^{k}$, (where $a$ is real, $b$ is
complex and $k=1$ $or$ $2$) are closed and periodic only for a discrete set of
parameter curves in the complex $b$-plane. It was further found that given
complex parameter $b$, the classical trajectories are periodic for a discrete
set of real energies (i.e. classical energy get discretized or quantized by
imposing the condition that trajectories are periodic and closed). Moreover, we
show that for real and positive energies (continuous), the classical
trajectories of $\textit{complex}$ Hamiltonian $H=p^{2}+\mu x^{4}, (\mu=\mu
_{r}e^{i\theta })$ are periodic when $\theta =4 tan^{-1}[(n/(2m+n))]$ for
$\forall $ $ n$ and $m\in \mathbb{Z}$.
