If it helps to think of them this way:
Take a fractal, often a Julia, then run it through a series of parameter changes, so your frac changes its shape, so you have a sequence of still images. Then, mentally 'connect the dots' of the outside edge of each still image. The quaternions are sort of the 3d shell that results when all those dots are connected.
Or to put it another way: If you were to take a cross-section slice of a quat, you would end up with a traditional fractal (often a Julia).
Many folks don't do them because they do take up so much computing power--essentially you have an near-infinite number of fractals being computed for just the one resulting pic.

RosemaryR
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"This...this is magnificent!"
"Oh, yeah. Ooooo. Aaaaah. That's how it starts.
Then, later, there's ...running. And....screaming."