The state of the art for arbitrary precision is in rapid flux right now. The naive approach is O(pixels * iterations * bits_precision^2) which is just hopeless. K. L. Martin[1] took it to O(iterations * bits_precision^2 + pixels * iterations) which was a massive improvement. I explain my webgl implementation of that algorithm here [2] and I think it's currently the fastest web implementation for depths up to 2000 bits. However, in 2022 Zhouran [3] dramatically improved the constant factor for the first term, then made a breakthrough in overall complexity, taking it to O(pixels * log(iterations) + iterations * bits_precision^2) which blows the approach I used out of the water for deep images. It's so recent that there's not really a blog post properly explaining it anywhere, you just have to sort through the forum thread., There is a webassembly implementation at https://fraktaler.mathr.co.uk/live/latest/ with public source.

1 http://www.science.eclipse.co.uk/sft_maths.pdf

2 https://www.hgreer.com/JavascriptMandelbrot/

3 https://fractalforums.org/fractal-mathematics-and-new-theori...