From what I understood, quaternion rotations could essentially be described as a vector coming from an object's location and going to (x,y,z) and W is a value that determines the object's rotation around that vector:
In this picture, the black arrow is the vector from the object's location (which I suppose would also be the origin in that object's local space?) to (1,1,1), my (x,y,z) point. The teal arrows circling the vector are essentially w, in that they represent some motion around the vector (they aren't a representation of magnitude though, since for some reason which I'm sure will be elaborated on, when w gets really high or low, it slows the rotation).
What I'm confused about is when I change (x,y,z) to (2,2,2) I have only changed the vector in magnitude, not direction. So I would think that the magnitude of the vector wouldn't effect wouldn't effect the rotation if W stays the same, but this shows otherwise:
I did some fiddling around and I found out that when (x,y,z) is (2,2,2) and w is 2, it gives the same rotation as when (x,y,z) is (1,1,1) and w is 1:
So I know, w=a and (x,y,z)=(a,a,a) gives the same rotation. I tried something else, comparing w=1 and (x,y,z)=(1,1,0.5) with w=2 and (x,y,z)=(2,2,1) and finally w=3 and (x,y,z)=(3,3,1.5) for good measure. They had the same rotation as well, so I know that as long as the four values maintain the same ratio, they will have the same rotation.
This where I've hit a bit of a standstill.