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Thanks!
Unfortunately shadows are quite tricky. For that short video I have already built a set of matte objects and a matte material to mask out the thing (pillars).
The matte object I have allows taking the shadows, but - as long as it's fairly simple to cast shadows on the floor and maybe a wall, it's getting tricker with more complicated objects.
That particular movie has a huge modern bench on the floor which makes it impossible (in practice) to take that approach.

In normal VFX pipelines, the way you'd normally integrate it is that you roughly model the scene (so add floors, walls, a rough model of the bench etc…) and just cast shadow on this geometry.
Not a big deal if you have a normal modelling tools, but impossible when you work with DE-based renderers and distance fields.

To make it more complicated, there are tracking imperfections and distortion of the backplate caused by the lens.
Again - something that is normally quite easy to correct for, but not in a 'primitive' environment for rendering fractals.

The only other workaround I see would be to render the fractal in any GLSLish shader and do all the compositing elsewhere.
But that would require to model a rough approximation of the fractal in a standard 3D modelling software to cast shadows.
(Or develop a pipeline to export voxels, convert to mesh etc…).

All would be doable on a hollywood movie, but not necessarily with the amount of time I plan on spending on these projects.