Suppose I have a system of complex ODE's of the form $$ i\dot{\mathbf{c}}(t)=\mathbf{f}(\mathbf{c}(t))$$ and I can write down a Hamiltonian such that each ODE can be written as $$\dot{c}_j=\frac{\partial\mathcal{H}}{\partial c_j^*}$$ for each $j$ where * denotes complex conjugate. As a very simple example, the Hamiltonian $$ \mathcal{H}=-i\left(|c_0|^2+|c_1|^2\right) $$ leads to the equations $$i\dot{c}_0=c_0, \qquad i\dot{c}_1=c_1. $$ Questions:
What are the conjugate momenta for this system (Are they just the complex conjugates)? Also, is there any way to transform this problem (via action-angle coordinates/madelung transform) to one where the Hamiltonian is purely real or where the system evolves under real dynamics? Is an imaginary Hamiltonian even an issue if I want to analyze a much more complicated non-linear system of this type using canonical perturbation or bifurcation theory?