The source info added by the OP identifies the fractal as a Julia set. It has evidently been distorted (warped) to give it a 3D appearance. I played around with a spiral-shaped Julia set I found at pixels.com and could create an image such as this. I'm unsure whether this is a viable route for the OP, if it is, probably some experimentation with the Julia set parameters will give a larger image to warp, so that it fits in a square box.
