What is the Madelung transformation and how is it used?

The Schrödinger equation is an axiom of quantum physics that is difficult to interpret i.a. because of its usage of complex numbers: $i\hbar \frac{\partial}{\partial t}\Psi = \mathcal{H} \Psi$.
In the Kopenhagen interpretation $\Psi\Psi^*$ is known to describe the probability distribution of a particle. Erwin Madelung tried (already in 1926) to understand the nature of the Schrödinger equation from a different point of view. He substituted $\Psi = ae^{i\phi}$ and split the Schrödinger equation in its real part and its imaginary part (first and second Madelung equation).
Because $\Psi\Psi^* = a^2$, the second Madelung equation (see above reference) becomes a continuity equation of the probability density $a^2$. This equation can thus be interpreted in a more physical way than the Schrödinger equation and that's the point. Similar arguments hold for the first equation but these arguments are more involved (see above reference).