Ordering Complex Numbers

Until now there has been no proper explanation of what the components of the complex number z = x + iy.
The best way of do this is to show how the argument (angle) and modulus (distance) derived from the x and y values have unique meaning (ordering).
The complex numbers are ordered not on a flat 2D plane like real numbers but in 3 dimensions.
The shape (or manifold) I use is called a zunnel and consists of a circle at one end morphing into a square at the other end.
Just as for a circle we need to fix its radius (that is the same as the modulus) and we also need to know the angle any point on the circle makes from a baseline or X axis.
The same is true throughout the zunnel but as the cross section changes shape we need to take account of this.
The zunnel actually defines the shape of the interior of a black hole! This is because Einstein's relativistic equations can be projected forward beyond light speed using my new complex mathematics.