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An antilinear endomorphism $J$ with square $-1$ equips a complex vector space with a quaternionic structure: complex multiplication supplies the quaternionic generator $i$ and $J$ supplies $j$, satisfying $Ji=-iJ$. Such a space has even complex dimension. There is no invariant complex line, because $Jv=cv$ would imply $-v=|c{|}^{2}v$. In dimension four the invariant complex two-planes are quaternionic lines, forming ${\mathbb{HP}}^1\cong S^4$. This is the Euclidean reality structure on twistor space used in twistor theory.

In flat twistor theory, a twistor is a pair $Z=({\omega }^A,{\pi }_{A^{\prime }})$ of two-component spinors. The twistor incidence relation with a spacetime point is ${\omega }^A=i{x}^{A{A}^{\prime }}{\pi }_{A^{\prime }}$. The nonzero vector $Z$ and its projective class play different roles: the latter belongs to projective twistor space.

Twistor theory encodes the conformal geometry of four-dimensional spacetime using incidence geometry in a complex vector space. In the flat correspondence, a spacetime point is a two-plane in a four-dimensional twistor space, and hence a projective line in projective twistor space. Null separation becomes intersection of these lines. Choosing a spacetime signature is an additional reality condition on the complex geometry.