Many years ago I wrote this fractal generator: http://uberto.fractovia.org/
It was shareware but then I put it as open source. It's written in Delphi, a language that I don't use anymore. So I'm playing lately with the idea of rewriting on some other lang (scala maybe?).
On the mathematical part I'm not expert at all, so I don't know if there is something a mathematician would find interesting. Maybe none because of software like Mathematica. I really don't know.
For the graphic oriented people I implemented some algos for continuous coloring, including the ones listed here:
Continuous coloring of a Mandelbrot fractal
but I'd like to add some new ones.
You might look to Mandelbrot's own work, a lot of which has to do with boundaries, also the relations of similar shapes on different scales and fractal dimension.