Fermion field can NOT be complex $\mathbb{C}$ or real $\mathbb{R}$. That is a common mistake in some text books. Fermion field must be real/complex/quaternion Grassmann variable.

Case in Point: the Majorana mass term would not be possible, if not for the Grassmann nature of the Fermion field.

In addition to real $\mathbb{R}$/complex $\mathbb{C}$/quaternion $\mathbb{H}$ Grassmann variable, you may go further up the division algebra ladder and toy with the idea of non-associative octonions $\mathbb{O}$ here

If that wets you appetite, you might want to check out sedenions $\mathbb{S}$ here.

Mind that one should be talking about octonion/sedenion **Grassmann** variables, the authors of the above linked papers failed to do so.

This post imported from StackExchange Physics at 2020-11-24 18:32 (UTC), posted by SE-user MadMax