Another two with this coloring; (2,2) rational function with parameters found by trial and error (using parameter animation in UF with 2D control surface).
Maybe once I understand this "matching" stuff Claude is doing it will become even prettier.
Can this "matching" be done visually with parameter animation?

You can also get very different results by changing the shape of the orbit trap. These three are the same Julia but with Point, Ring and Cross shaped traps. Oh, and triangle average as a separate layer. The Bubble colouring is used with the blue palette and Halo with red.

I have posted a few more in my image thread: https://fractalforums.org/image-threads/25/experiments-with-colouring-and-layers/1231/315

If you let the decIter variable count reductions of the 2nd closest distance, too, like so:

    // sort current distance into our "list" of closest and 2nd closest
    if (trap < min) {
      min2 = min;
      min = trap;
      mIter = iter;
      ++decIter;  // number of steps to reach minimum distance
    } else if (trap < min2) {
      min2 = trap;
      ++decIter;  // EXPERIMENTAL: also count secondary reductions
    }

then the insides can get really busy.

My intuition fails me as to what the additional boundary curves are connecting or separating. Basically, the inside regions just shatter into (too?) many little slivers, with each of the original cells being subdivided further.

At first I thought I wanted the insides to be more and more detailed. But more experiments didn't yield anything that seemed to be "real"; i.e. nothing that tickled my mind in the specific way to make me believe there might be some mathematical structure to be discovered.

I rendered further surveys, and some of them looked nice. But that, too, did not seem to be going anywhere interesting.

Now I am systematically mapping the influence of the orbit trap position on the resulting appearance. Or at least, somewhat systematically. Because there are still too many arbitrary "design parameters". Nonetheless, structures did emerge from the fog, so to speak.

In the following image, for each single pixel, four entire Mandelbrot images have been computed. The pixel position determines an orbit trap position, and the four Mandelbrots are rendered with a minimally moved orbit trap position. Essentially, I am trying to compute the rate of change (of the corresponding Mandelbrot "cells") around "all" possible orbit trap positions.

Dark pixels mean slow change, bright pixels mean fast change, as the orbit trap position moves around that area.

Hmm. Those atlases might just be inefficiently and incorrectly rendered wrong buddhabrots.

... and animated:
http://vectorizer.org/3doodle/BlackWalled001.mp4

This is exactly the sort of content I was hoping to find by coming to this site! I love this approach for coloring, for a few reasons... firstly, it looks great. And secondly, it can teach me new things about some very familiar shapes...

I set this idea up as a coloring method in my own fractal toy and came up with some pretty good results. I made the cell boundaries black instead of white, but it's the same idea. I did not find a very good way to get highlights... Any hints available for how to calculate those?

I'll attach some pictures. First one will be a pretty standard Julia (not very deep) with this coloring method.
And the next one will show the Burning Ship fractal, because I love how it gives some surprisingly structured detail to a section of this fractal that I had always assumed was boring! (The bottom right corner of the shape)

And finally I'll post a zoomed-in view of that bottom corner, so we can all enjoy the chaos that resulted.

Edit: Trying again to get the images to show up properly