I'm wondering if the concept of "divisibility" also works in complex numbers.
For example, if I say $a+bi$ is divisible by $c+di$. Does this mean that there exists another complex number $\alpha+\beta i$ such that $$\begin{align*} (c+di)(\alpha+\beta i)=a+bi \end{align*}$$If this is true, can I say that $5i$ is divisible by $5$ and is divisible by $i$ since $$\begin{align*} 5i=(5+0i)(0+i) \end{align*}$$
You certainly can do this. You can do this in any situation where multiplication makes sense -- although you have to watch out that in other settings, divisibility might not work the way you're used to.
If you use this definition for all complex numbers, you get something very boring: every complex number $z$ is divisible by every non-zero complex number $a$ because $$ z = \frac za \times a. $$ (You get something equally boring if you use all real numbers.)
However, if you limit yourself to complex numbers $a + bi$ where $a, b$ are integers, you get the Gaussian integers, and there divisiblity is certainly interesting.