How can complex polynomials be represented? I know that real polynomials (polynomials with real coefficients) are sometimes graphed on a 3D complex space ($x=a, y=b, z=f(a+bi)$), but how are polynomials like $(1+2i)x^2+(3+4i)x+7$ represented?
 A: For good ways to visualize many things related to complex variables, take a look at this wonderful book by Tristan Needham.
If your "$x$" is real, then $x \mapsto (1+2i)x^2 + (3+4i)x + 7$ is a mapping from the real numbers to the complex plane. In other words, it's a parametric curve in the complex plane, with $x$ being the parameter.
If your "$x$" is complex, then $x \mapsto (1+2i)x^2 + (3+4i)x + 7$ is a mapping from the complex plane to itself, which can often be visualized as a deformation of some kind.
Anyway, look at the book. It's very good.
A: This is a $2 \times 2$ graph, it can't be done "completely". But you can show some "slices" of the data, e.g. the real and complex components along the real line (or some other line of interest), or perhaps magnitude and angle. Or do something like a velocity field. It depends on what is to be shown.
Whatever is done, a graph is a visual rendering of something, designed to make people understand. Always keep that in mind.
A: Plotting $P(z) $ and $P:\mathbb{C}\rightarrow\mathbb{C}$, if you want to use a cartesian coordinate system give a 4D image because the input can be rappresented by the coordinate $(a,b)$ where $z=a+bi$, and the output too. 
So the points (coordinates) of the graph are of the kind 
$$(a,b,Re(P(a+bi)),Im(P(a+bi)))$$ 
and they belong to the set $\mathbb{R}^4$ (set of 4-uple of $\mathbb{R}$).
I think is impossible for evry human to watch a 4D plot on paper, but if you do a 3D plot with an animation, the timeline is your 4th dimension, and you can see how the 3d plot changes when the time (4th variable) changes.
I hope that is correct. At least this is how i represent in my mind 4D objects.
ps:I apologize in advance for any mistakes in my English, I hope that the translator did a good job :).
