What subset of physics can be described in complex phase space? By complex phase space I mean $z=q + i\,p \in \mathbb{C}^{n}$ and  $z = q -i \, p \in \mathbb{C}^{n}$ and the Hamiltonian $H(z,\bar{z})$ is a function of $z, \bar{z} \in \mathbb{C}^{n}$.
 A: What follows doesn't completely answer your question but hopefully it can be useful. 
A classical example of an hamiltonian $H(p,q)$ which can be reduced to the form $H(z_+,z_-)$ with $z_{\pm}=\alpha q \pm i \beta p$ is the harmonic oscillator. The hamiltonian of an harmonic oscillator is
$$H(p,q)=\frac 1 {2m} [p^2 + m \omega q^2]$$
Using the identity
$$p^2+m \omega q^2 = (p+i q \sqrt{m \omega})(p-i q \sqrt{m \omega})$$
we see that we can write the hamiltonian as
$$H(p,q) = \frac 1 {2m}  (p+i q \sqrt{m \omega})(p-i q \sqrt{m \omega})\\= \left(\frac p {\sqrt{2m}} + i  q  \sqrt{\frac \omega {2}}\right) \left(\frac p {\sqrt{2m}} - i  q  \sqrt{\frac \omega {2}}\right)$$
and defining
$$z_{\pm} = \left(\frac p {\sqrt{2m}} \pm i  q  \sqrt{\frac \omega {2}}\right)$$
we obtain
$$H(z_+,z_-) = z_+ z_-$$
The same trick can be used whenever $p,q$ appear as couples of quadratic terms in the hamiltonian. In quantum mechanics, this trick leads to the definition of the creation and annihilation operators.
