Quaternionic structure on a complex vector space ($J^2=-1$)

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.