Any complex number $z=a+ib$ can be represented as a point $(a, b)$ on the Argand diagram.
Find some complex numbers which square to a real number. Can you find all such complex numbers? Do all real numbers result from such a square? Represent your findings on the Argand diagram.
Find some complex numbers which square to an imaginary number. Can you find all such complex numbers? Do all imaginary numbers result from such a square? Represent your findings on the Argand diagram.
Explore the effects of squaring on other complex numbers as they are represented on the Argand diagram. In particular, can you find any complex numbers which square to the complex numbers found in the first two parts of the problem?
Explore further any conjectures or questions which arise.