The programs I found to render Mandelbulbs are all working directly on the graphics card. But my plan was to use the result in Blender and for 3D prints so I needed the triangles e.g. as STL. Finally I came up writing my own Mandelbulb program. It is not as fancy as many other programs with shaders running on the graphics card, but it does what I wanted: Export a section of the Mandelbulb as STL, render it and 3D print it.
TeasersResult of rendering the resulting STL in Blender as SSS material:
Result of detail rendering the resulting STL in Blender as resin-like material: