"The original Mandelbrot is an amazing object that has captured the public's imagination for 30 years. It's found by following a relatively simple math formula. But in the end, it's still only 2D and flat - there's no depth, shadows, perspective, or light sourcing. What we have featured in this article is a potential 3D version of the same fractal."
So, the 3-D version looks in places like organisms, in places like weather, in places like geography. But sitting in a dark house at 3am staring into a zoom it feels extremely freaky.
The link above is to a discussion and some still shots but there are animated zooms.
Here's a great animation of the Julia Set realised in 3-D. I think this is a good place to start. It's like entering into an alien life form.
This one from Youtube is a small glimpse, but bear in mind it's only in orthographic mode, so there's no tasty perspective or parallax as yet. In other words, it's like enlarging a giant photograph, rather than flying through - plane-style. Watch this space for future animations. Also, if you want to make any yourself, and would like them to featured on this page, let me know, or even better, post to the thread at FractalForums.com.
Also check out David Makin's animations, such as the degree 4 version of the Mandelbulb, his Crater Lake Flyover, Krzysztof Marczak's excellent rotational Mandelbulb variation, and this computer generated romanesco broccoli created by Aleksandar Rodic (which is an IFS type fractal rather than the Mandelbulb we're exploring, but still really cool).