Hopefully somebody understands what I mean here,
If take a polynomial with complex numbers as input, then I will get a complex number as an output. If the input and output are plotted on an Argand diagram geometrically how do quadratics, cubics ,quartics etc.. transform the points?
I'm interested to know what this area of Mathematics is called or where I can read about the theory?
Kind regards, Cliff
On
If there is a name that can be given to this field of study, it is conformal mappings (http://mathworld.wolfram.com/ConformalMapping.html), this name being associated with the fact that angles are preserved (even is this kind of transformations, in certain cases, do not use complex numbers).
One should set apart transformations using first degree polynomials. More precisely $z \rightarrow Z=az+b \ (a \neq 1)$ is a well known transformation : the similitude with angle arg(a) and ratio $|a|$, the center of the similitude being obtained by solving the fixed point equation : $z_0=az_0+b$ i.e., $z_0=\dfrac{b}{1-a}$.
Remark: if $a=1$, we have the translation by vector $(Re(b),Im(b)$).
The set of transformations $z \rightarrow Z=az+b$ without restriction on $a$ constitute a group for function's composition called the homothety-translation group.
Above degree one, nothing general can be said, but nice things can occur.
For example, $z \rightarrow Z=z^2$ (see figure) transforms a rectangular grid into a grid made of a double "net" of mutually orthogonal parabolas (preservation of right angles) with a common focus (the origin), the blue and red colors helping in the understanding of "what is transformed into what".
For example vertical line which is the set of $z=1+it \ (t \in \mathbb{R})$ is bended to form a parabola described as the set of complex numbers of the form $Z=z^2=(1-t^2)+i(2t)$ which is the parabola passing through $1, \pm 2i$.
Edit: conformal transformations with polynomials say of degree $>2$ are far from being the most interesting. Among the very used/useful transforms :
homographic transforms $Z=\dfrac{az+b}{cz+d}$, essential in hyperbolic geometry. Among them $Z=\dfrac{z+1}{z-1}$ (Poincaré transform) which maps in a bijective way the open left plane (x<0) onto the open unit disk.
almost all functions linked to classical functions, $Z=\exp(z)$, $Z=\log(z)...$.
transforms with fractional powers like $Z=\sqrt{z}$ that bends cartesian grid lines into hyperbolas. You can try also $Z=\frac{1}{\sqrt{1-z^2}}$. Using these kind of transforms + the concept of integral of a complex function, one can map for example everything into a unit square ; this has deep implications in the so-called "elliptic functions" (see Schwarz Christoffel transforms, with applications to hydrodynamics in www.me.unm.edu/~kalmoth/ME530-ch4.pdf).
Joukowsky transform $Z=z+\frac{1}{z}$ https://en.wikipedia.org/wiki/Joukowsky_transform (also with applications to hydrodynamics).
etc.
All this is part of one of the most beautiful part of mathematics, complex functions theory.
You're probably asking how to draw the graph of polynomials with inputs from the complex plane. Well the graph is clearly 4-dimensional, so the best we can do is to use two graphs. You can try Wolfram Alpha to see one possible way these things are graphed. Your "transform the points" is probably the same idea as what WA depicts as the "complex map".