Consider the complex plane. We can understand it as a two-dimensional real vector space. We have a two-valued pseudo-map, $z \mapsto \sqrt{z}$, on that plane.
Take a nonzero complex number. Consider one of its images under that map. Naively, it's another complex vector. Rotate the domain plane once through a full turn around the origin. i.e. 360 degrees. The value of the square root now becomes its negative. Rotate the plane again once around the origin, i.e. a total of 720 degrees. The value of the square root now is back to where it was.
But this sounds very much like the idea of a "spinor" from quantum physics. Is it thus possible to consider, in addition to the "classical mathematical" understandings of $z \mapsto \sqrt{z}$, as a "multi-valued function", or as a function mapping out of a Riemann surface, or as an arbitrarily-choosable branch of a multi-valued function, that instead it is a true function with domain $\mathbb{C}$ that, instead of returning a complex number, returns a spinor? Or if not that, then in some other way, we can connect it to spinors?
Moreover, consider $z \mapsto z^{1/n}$ for integer $n > 2$. Can we generalize the idea of spinors, then, to "$n$-spinors" which require $n$ rotations of the base space to turn them back to what they originally were?
The reason I'm thinking of this is the idea of spinors seems rather challenging pedagogically, and it'd be great if one could easily introduce it with such a simple example or at least in a small number of steps from building on this simple example.