Comment by trylist

Clifford algebra is exactly how I figured it out as well.

Here's the clearest explanation I can make:

Things have a "sided-ness" to them. In most mathematics, the definition of a plane ignores this sided-ness. Three points and you have a plane, right? But you can't describe the orientation of a plane with the points, because you can't tell which side is "up". A piece of paper has two sides, why doesn't a plane? This is important when describing rotation!

Anyone who has used a look_at function can understand this. You give it a point for the eye position, and a point for the eye to "look at". What is not described is how that eye is oriented. Is it upside down? Sideways? Right side up? Even vectors have a "side"! This is the "twist" that is often mentioned.

If you imagine yourself as the eye, you can imagine your eyes being in the same position, looking at the exact same point, but in many different perspectives. Lying on your side, doing a headstand, standing upright.

I'm not an expert and I'm probably wrong, but intuitively this is what I think that fourth dimension on the quaternion corresponds to.