[NAFV_P]

  z = x + i*y = e^(a + i*b) = e^a*(e^(i*b)) = e^a(cos(b)+i*sin(b))

Both sin and cos are many to one functions. In the equation above, replacing

  b

with

  b + 2*pi*t

where t is any positive or negative integer, would result in the same complex number.

[NaNaN]

We've used the same number for those different rotations. That does not do anything directly about irrational power p.

If p is a rational number represented by m/n, then

    360 / (m/n) * m = 360n

So there are multiples of 360 that are divisible by m/n, and we can get a unique complex number from z^p.

Now p is irrational, what happens? According to what you said, there are so many different graphs that can not be merged into just one.

[NaNaN]

EDIT: So ... we can get FINITE complex numbers (or graphs) from z^p.

Now p is irrational, what happens? According to what you said, there are infinite different graphs that can not be merged into just SOME.

[NAFV_P]

I am replying to the last paragraph of darsham's comment.