See for example the works of Laurent Nottale:

https://en.wikipedia.org/wiki/Laurent_Nottale

Relayed ideas have been advanced by others.

In his book Mandelbrot mentions “a new fractal wrinkle to the presentation of quantum mechanics.”

“Feynman & Hibbs 1965 notes that the typical path of a quantum mechanical particle is continuous and nondifferentiable, and many authors observe similarities between Brownian and quantum-mechanical motions (see, for example, Nelson 1966 and references herein). Inspired by these parallels and by my early Essays, Abbot & Wise 1980 shows that the observed path of a particle in quantum mechanics is a fractal curve with D=2. The analogy is interesting, at least pedagogically.”

Perhaps the analogy is more than pedagogical. The similarity between quantum paths and fractals seems to indicate that fundamental reality itself has a fractal structure.

I'm wondering how these ideas relate to yours.

Best - GP