 Complexified Minkowski spacetime

Allow the four affine Minkowski coordinates to take independent complex values, obtaining $M_{\mathbb{C}}\simeq {\mathbb{C}}^4$ with the holomorphic extension of the flat conformal metric. In two-spinor coordinates $x^{A{A}^{\prime }}$, the metric is proportional to ${ϵ}_{AB}{ϵ}_{A^{\prime }{B}^{\prime }}d{x}^{A{A}^{\prime }}d{x}^{B{B}^{\prime }}$. The twistor incidence relation associates a complex projective line to each affine point; compactification adds the points at conformal infinity.