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I'm not sure if this answers your actual question, but you might be interested in code that Simon King and I have at # 11837 for generating Newton basin fractals. You are right that it is slow. Hopefully not quite as slow as yours, though, maybe it's all you need.

Incidentally, even at David Joyce's awesome generator page he just takes the roots; trying to get that from a generic polynomial first will severely slow you down, I'm sure.