So, I understand complex number multiplication, and how it represents $2D$ rotations.
What I don't understand is, how you add two more imaginary numbers $j$ and $k$, and get $3D$ rotations. I believe Hamilton "made up rules" on how to add these extra imaginary numbers, but how did he construct it in a way that worked?
I also don't understand the purpose of adding two imaginary numbers instead of just $1$ more for $3D$ rotations.
In this paragraph, I will explain my attempt to visualize / construct the quaternion, you don't have to read this, it's probably utterly wrong. I tried to imagine a $3D$ space with the axis vectors $1$ (the real number line, side to side), $i$ (up), and $j$ (depth). Next, I imagined a vector pointing out from the origin with a real component and a $j$ component. Next, I imagined multiplying this vector by $i$ to rotate it $90$ degrees up, so I would get a resultant vector with an '$i$' component and an '$j$' component. The problem is, multiplying (real + $j$) with ($i$) algebraically gives you an '$i$' component and an '$ij$' component, which does not make sense to me. This is where I got stuck.