Oscillator Differentials: What's a physical meaning of complex part of the solution for coordinate change of the anharmonic oscillator?

[DifferentialGalois]

Homework Statement

What's a physical meaning of, for example, complex part of the solution for coordinate change of the anharmonic oscillator?
Why after substitute (for diff. equation solve) for real x we can earn $x = Re(z) + iIm(z)$? Is it because of substitutio?

Relevant Equations

$x = Re(z) + iIm(z)$
$x(t)=e^{i\alpha t}$
$x(t) = A e^{i \alpha_1 t} + B e^{i \alpha_2 t}$

I don't understand what the question means, and the answer is provided here: https://physics.stackexchange.com/a/35821/222321
Could someone provide a comprehensive one-by-one explanation.

 

[haruspex]

The discussion at that link does not, as far as I can see, provide a physical meaning to the complex solution. Neither does it address anharmonicity, so I assume you mean just a standard damped (maybe forced) oscillator.
If we map the state onto the complex plane, the graph as a function of time (an axis normal to that plane) becomes a helix, tapering exponentially in the case of unforced. I would think this could be realised in a physical system.

 

[DifferentialGalois]

bump

 

[berkeman]

bump

Why are you bumping your thread and not replying to @haruspex ?

 

[DifferentialGalois]

Why are you bumping your thread and not replying to @haruspex ?

i need an explanation to the mathematical equations.

 

[berkeman]

i need an explanation to the mathematical equations.

I thought I saw a pretty good explanation in the post by @haruspex -- Which part of what he wrote did you not understand?