You are heading into deep territory! Discrete groups of PSL(2,C) are Kleinian groups, and I recommend Marden's book "Hyperbolic Manifolds". http://amzn.to/2H2Wr6i
@roice713 what happens when you compose these discretely: do a loxodromic, do a parabolic, and repeat? (I can't quite see that we get the same limit sets as using circles and reflections)
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The answer will depend on the loxodromic and parabolic you choose. In a simpler case, you could choose the parabolic fixed point to coincide with one of the loxodromic fixed points. In that case, I think the limit set will still just be 2 points.
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