This is, I think, the difference between thinking of a quaternion (or a vector!) as a collection of numbers with rules for manipulating them, and thinking of a

This is, I think, the difference between thinking of a quaternion (or a vector!) as a collection of numbers with rules for manipulating them, and thinking of a quaternion (or a vector) as an abstract geometrical object that exists in a space and has certain properties. The fascinating thing, to me, is that for so many geometrical objects, you can start with a short list of straightforward properties, and show that they exactly correspond to the collection-of-numbers concept over an arbitrary basis. Oftentimes, if you want to do something to the stuff on your screen, if you can describe it in terms of the abstract objects with no reference to the basis, you can then transliterate that to operations on a bunch of floats in a way that is simpler and more robust than working with the floats from the beginning.