Quaternions are difficult to visualize because they exist in a four dimensional hypersphere, S^3 beyond our full perception. Mainly for this reason, they have not been in the regular science curriculum for a hundred years. To avoid gimbal lock in 3D rotations, quaternions came back into favour with the development of computer graphics.
It is therefore not that much of a surprise that quaternions describe quantum hyper-helicity and are elements of reality in Nature. That means that Nature has dimensions beyond our normal spacetime. Quaternions are real objects and are responsible for the spinning of a spin axis. Since this occurs in a 4th spatial dimension, and all we can see is the stereographic projection onto our spacetime, our view is restricted to the two polarized states of up and dn. We cannot observe the other two hyper-states which are the left and right helicity of a spin.
The only reason to write this blog is to refer you to a set of videos with excellent graphics which explain quaternions and give a clear view of the stereographic projections ( that what we can see).
Here is an overview of quaternions and their projection
Here is a series of lessons that visualize quaternions in detail.
You can find a lot more at 3Blue1Brown site
Spin has 4 dimensions but all we see is the vector projection of a unit quaternion into our 3 spatial dimensions, and that projection is our two state spin described by Dirac.
