quantum mechanics - Schrodinger equation contains two pieces of information?

I've read that Schrödinger equation contains two pieces of information: about amplitudes and about phases. I know continuity eqn. for probability which is nice expression of former. I wanted to somehow isolate the latter. My idea was the following (I consider single particle in potential $V$ that only depends on position and neglect spin - so state is represented by one complex wavefunction). I use following representation of wave function:

$\Psi=Ae^{iS}$

In following amplitude $A$ is nonnegative and phase $S$ is real. For stationary states I got following set of equations equivalent to complex stationary S. Eqn. (to which I shall refer as 1. and 2. respectively):

$\Delta A+2A(E-V)-A(\nabla S )^2=0$

$A \Delta S +2 \nabla A \cdot \nabla S =0$

Where I clearly put some dimensionful constants equal to $1$ for simplicity. For general states I got the following, equivalent to S. Eqn. (eqns. 3. and 4.)

$\Delta A = A(2V+ (\nabla S)^2 + 2 \partial _t S)$

$A \Delta S + 2 \nabla A \cdot \nabla S +2 \partial _t A =0$

If you can answer any of the following questions, it would please me greatly.

Do these equations taken just one at the time admit any straightforward interpretation?

Can they be decoupled in any way? I suppose not, but maybe in some special cases...

Are they actually useful or is it always more convenient to not bother and keep a complex wave function?

Is there some other way to isolate the information about the phase? I've been able to derive some other relations using complex representation, but they didn't look interesting or useful.

Answer

There is an existing subfield of physics that does exactly this, the de Broglie Bohm (dBB) version of quantum mechanics.

This phase information can be expressed to look like the classical Hamilton-Jacobi equation for a parameterized family of initial value problems except the classical potential has an extra term that depends on what you call A. It is called the quantum potential.

So an obvious interpretation (explicitly made by the dBB interpretation) is that what the wave function does is create an additional potential that exerts additional forces on top of the classical forces (the classical forces being the ones that arise from the regular, classical, potential). All in a way that reproduces the predictions of quantum mechanics.