Is there a way to visualize, on the complex plane, complex numbers multiplication? Easy question today (i suppose).
As title said, is it possible to visualize multiplications between two complex numbers over the complex plane?
 A: Use polar decomposition: then we are multiplying everything by $re^{i\theta}$. This can be carried out in two stages, (i) multiplication by $r$ then (ii) multiplication by $e^{i\theta}$. 
In the first stage every complex number (regarding $x+iy$ it as a vector from origin joining $(x,y)$ ) geometrically  gets stretched by the factor of $r$ (or shrunk, if $r<1$).
In the second stage geometrically every number rotates around the origin by the angle $\theta$.
So a complex number multiplication involves these two geometrical transformations (in any order).
A: Somewhat, you have of course include the coordinate system in some way in the visualization.
Suppose you have two complex numbers $A$, $B$ and their product $C$ as points and the oringin O and unit $U$ (ie $1$). Then you can construct $C$ as constructing a triangle similar to $OUA$, but with the side $OB$ in place of $OU$. If that triangle is oriented the same way as $OUA$ it will be $OBC$.
Using complex arithmetics it's straight forward to see that this is correct.
