A fairly recent technique is to use perturbation theory to approximate the solution to the complex equation for points near a decently chosen starting point. The approximation massively reduces the amount of slow big decimal arithmetic needed for deep zooms, while retaining accuracy. Without getting into the actual math being perturbation theory, http://superfractalthing.co.nf/sft_maths.pdf is a good start for just dumping some calculations into your fractal viewer to get good results.

I think this is a youtube upload of one of the animations from http://www.hpdz.net/ -- there's a ton others on there too.

They go into the techniques a bit on the site under the "technical" section.

I remember seeing seemingly endless fractal zoom animations like these as long as 20 years ago. Basically, you're not calculating the entire original field of view to anything near the final level of depth. At any given frame, you're essentially doing the calculations only for what's in the current frame and only to the level of detail to support the current pixel count.

Well, yeah, but the mantissa alone on your numbers is going to get ridiculously long. That's going to take a bite out of your CPU even just for the ~1000x1000 pixels in the frame

> Well, yeah, but the mantissa alone on your numbers is going to get ridiculously long. That's going to take a bite out of your CPU even just for the ~1000x1000 pixels in the frame

Utterly devastate the computational ability of your CPU. I remember setting up my and k6 300mhz to compute stuff sometimes for days to get awesome fractal deep zooms like this. The computation is embarrassingly parallel but no hardware handles numbers that precise with any reasonable speed. Something like 1024 bit floating point in fractint back in the day. I'd love to see an efficient GPU implementation of that kind of stuff