What does it mean to divide a complex number by another complex number?

Complex numbers' multiplication is better understood if you forget the "cartesian vector in the complex plane" analogy: $$ z = a + b i \quad z \in \mathbb{C}, a, b \in \mathbb{R} $$ And in stead think in polar coordinates: $$z = r \angle \theta = r e^{i \theta} \quad z \in \mathbb{C}, r \in \mathbb{R}^+, \theta \in [0, 2 \pi) $$ Wherein $r$ is the magnitude, $\theta$ is the angle.

When multiplying it is easy: $$ z w = (r_z r_w) \angle (\theta _z + \theta _w) $$ You add the angles and multiply the magnitudes.

When dividing you do what comes naturally: $$ \frac z w = \left(\! \frac{r_z}{r_w} \!\right) \angle (\theta _z - \theta _w) $$ To divide means to find the difference in angles and the factor in magnitude.

It means to find another complex number $y$, such that $xy=w$. (Just as it does for real numbers.)