It seems completely natural that going from 2d to 3d adds a new dimension of awesomeness to fractals. Accordingly, it is quite a shame that until very recently, I was not aware of the magnificent Mandelbulb. If this is a new world for you, too, go and explore the wonders of 3d fractals using tools like Mandelbulb 3d.
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Glimpses of a fascinating world 
What about 4d now? Did you ever think fractals could be used in rendering art digitally? http://www.segmation.com
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In 4d, defining the analogues of the Mandelbulb fractal is somewhat more straightforward because we have a division algebra structure there in form of the quaternions. However, this seems to lead to much smoother, less fractal, fractals. Of course, fractals (2d/3d/4d/…) are a mainstay of digital art; see many examples on platforms like DeviantArt.
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