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  I posted a few images over in the images forum when I discovered the formula.  For z^2, it has a central spin hole where the main cardioid is in the 2d Mandelbrot:


although it doesn't include the 2d Mandelbrot cross section, as it's more closely related to a burning ship fractal.

  It (the z^2) has little mini-swirls all throughout the body of the fractal, and some crystalline sections like:


  And then there are the sections in which it is all twisted around itself:


another zoom into the twistyness of the z^2:


  That is just the z^2.  The z^3 has interesting properties of its own, as does the z^4 (although I'd think that the properties of the z^3 and 4 are repeated through all the higher n(s), just different basic patterns). 

z^3 has these neat interconnected X things (well, they are more like an X and a + sign melded together):


And z^4.. well it has cool buildings, moats, and just some generally complicated patterns:

First zoom into buildings:

going towards 'em:

iterated once:

between the buildings (moat area between complexes of buildings), deep zoomed (well, 2 more iterations from above.. not tooo deep):



  So there is lots of need stuff... anyways.