In this link http://benasque.org/2009gph/talks_contr/074Herdeiro.pdf page 15, it was said that:
"Use spinorial geometry techniques: One takes the space of Dirac spinors to be the space of complexified forms on $\mathbb{R}^2$, which is spanned over $\mathbb{C}$ by {${1, e_1,e_2,e_{12}}$}."
Though this has to do a little with physics but mainly the requirements are from mathematics. Why is it that we can take the space of Dirac spinors to be the space of complexified forms? And why on $\mathbb{R}^2$ which is spanned over $\mathbb{C}$ in this case?