In this video lecture about mathematical physics (after 1:20:00) Carl Bender explains that allowing the energy of a complex classical oscillator to become imaginary can some kind of mimic quantum behavior of the system.

To explain this, he uses the example of a classical particle with the potential energy given by

\(V(x) = x^4 -x^2\)

If the energy of the particle is larger than the energy of the ground state (but smaller than the barrier between the two potential wells) $E_1 > E_0$ it can oscillate in one of the two potential wells centereed around the two minima of $V(x)$, but not travel between them.

However, allowing the energy of the particle to become imaginary, say

\(E = E_0 + \varepsilon i\)

the trajectories of the particle in the complex plane are no longer closed and it can some kind of "tunnel" between the two potential wells. The "tunneling time" $T$ is related to the imaginary part of the energy as

\(T \sim \frac{1}{\varepsilon}\)

In addition, Carl Bender mentions that taking the limit $\varepsilon \rightarrow 0$ of such a complex oscillator system some kind of mimics the process of taking the classical limit of a quantum system.

What are the limitations of this analogy between a complex oscillator with imaginary energy and a "real" quantum system (no pun intended ;-) ...)?

An aside: the jumping between the two potential wells of a complex oscillator with imaginary energy reminded me of a Lorenz Attractor ...