Decided to take a look at what the derivative-space of z_n looked like and ended up finding this. Inspired by the derivative bailout method (aka derbail) the usual "Buddhabrot projection" of the ZrZi space is replaced with a projection where the pixel positions are determined by the complex valued derivatives instead. The value of c is set to around -0.660+0.331i which is a period-9 tangent point where the Mandelbrot main cardioid sprouts a smaller bulb corresponding to an internal angle of 160° or four ninths of a turn. The initial z₀ values are random sampled from a small disk centered at roughly 1.055-0.227i — a smaller sub-region within the Julia set of the chosen c.

For all intents and purposes all the resulting orbits are stable, although to a varying degree. For the most stable orbits the iterative derivatives quickly shrink to zero, but the closer the initial z₀ value is to the Julia set boundary the longer it takes for the derivative to settle down to zero. The image above shows the relative density of those derivatives which unsurprisingly display a periodicity corresponding to the orbit origin. I have yet to find out why the bright point-like overdensities are so neatly discretized or what their tantalizing overall pattern signifies.