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Nice!

Well, this formula is more a "mandelbrot-esque" one indeed. The original formula Z=Z+1/Z+C, setting initial Z value to 1, gives a strange Mandelbrot, without even using any power function (Ok, 1/Z is equal to Z^-1, but...) My finding is the results of adding a negative scale factor. It's all-inside, it doesn't diverge, but there's no folding so no Kaliset-esque patterns here. You can add abs function for that. But I think we shall call the patterns "Duck-esque" as it was Sam Monnier's formula which made them popular first

Awesome fractal, very nice

A bit of experimentation got me the attached image in short order. There's a minibrot inside a spiral, colored by calculating the Lyapunov exponent of the moduli of the orbit values attained in the first 1000 iterations. The critical point iterated was 1.

Notable mathematical features: First, in the dynamic plane, zero is the only finite preimage of infinity. Infinity is a fixed point which is repelling if |scale| < 1, indifferent if |scale| = 1, and attracting if |scale| > 1. So, there are escaping points if the scale is positive. It's never superattracting, and the shape of its basin can be selected by varying the argument of scale.

Further, if |scale| < 1, there are generally chunks of the c-plane near c = 0 where there are no attractors at all -- all periodic points are unstable and the Julia set is the whole complex plane. In the rest of the c-plane one or both critical points find finite attractors, sometimes finding two separate ones. A coloring method like exponential smoothing or Lyapunov can reveal the dendrite structures. In the attached image, the spirals are the dendrite structures, which are largely space-filling here except in the minibrot, an island of stability where the critical point 1 is associated with a bifurcating attractor.

The normal Mandelbrot set has a "default". Infinity is always attracting, and is itself a critical point that is captured by itself. The other critical point, zero, can go there or to a finite attractor. A rule of rational maps of the complex plane is that every attractor must capture at least one critical point; so there is no attractor other than infinity unless zero finds it. So, the fate of zero tells you everything about the dynamics. Moreover, though, when zero isn't on the Mandelbrot figure (going to a finite attractor) or on its boundary, including on a dendrite, then it must go to infinity. If infinity disappeared as an attractor, the only choices for zero would be finite attractor or dendrite. So the space between the M-set's dendrites, where zero goes to infinity, would have to disappear and the dendrites would have to expand to fill all of that space. That is what has happened here: whereas a normal seahorse spiral has a clear space between the spiral arms, here the spiral arms are crammed up against one another without any room between them, because every point not in a minibrot must be on a dendrite here.