Geometric interpretation of the multiplication of complex numbers?

Suppose we multiply the complex numbers $z_1$ and $z_2$. If these numbers are written in the polar form as $r_1 e^{i \theta_1}$ and $r_2 e^{i \theta_2}$, the product will be $r_1 r_2 e^{i (\theta_1 + \theta_2)}$. Equivalently, we are stretching the first complex number $z_1$ by a factor equal to the magnitude of the second complex number $z_2$ and then rotating the stretched $z_1$ counter-clockwise by an angle $\theta_2$ to arrive at the product. There are several websites that expand upon this intuition with graphics and more explanation. See this site for example - http://www.suitcaseofdreams.net/Geometric_multiplication.htm

Add the angles and multiply the lengths.

Yes, there is a simple geometric meaning, but you need to convert to the polar form of the complex numbers to see it clearly. $3+i$ has magnitude $\sqrt{10}$ and angle about $18^\circ$; $-2+3i$ has magnitude $\sqrt{13}$ and angle about $124^\circ$. Multiplication of the complex numbers multiplies the two magnitudes, resulting in $\sqrt{130}$, and adds the two angles, $142^\circ$. In other words, you can view the second number as scaling and rotating the first (or the first scaling and rotating the second).