I was reading this paper: http://dinamico2.unibg.it/recami/erasmo docs/SomeOld/RevisitingSLTsLNC1982.pdf It is on superluminal Lorentz transformations and is too advanced for me. But anyway, take a look at equation(s) (11). For the y' and z' transformations, there is an imaginary unit coefficient. But for the x' transformation, there is none. So the space in the x-direction is 'real', but the transverse space around it is 'imaginary'?