Let's say I have a complex number, $z=a+bi$ such that $z$ is a root of unity. It's clear that $\mathrm {arg}(z)= \arctan\left(\frac{b}{a}\right)$. Can I conclude that $z$ has $\left|\frac{2\pi}{\mathrm {arg}(z)}\right|$ distinct integral powers?
What can we say if I raise $z$ to real powers rather than integral? And lastly what about complex powers? Will there be finite distinct powers?
The only case in which there are only finitely many distinct positive integer powers $z^n$ is when $z$ is a root of unity. When non-integer powers are allowed, there are always infinitely many.