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Here are a few illustrations of some simple complex functions, produced using Maple:

Function	Animation
z → z + (1+i)	view
z → z × (1+i)	view
z → z + 1/(1+i)	view
z → z × 1/(1+i)	view
z → z2	view1   view2
z → z3	view
z → ez	view1   view2
z → sin(z)	view1   view2