This is a useful way to think about it, but you have to keep in mind that (even flat) spacetime is not Euclidean but Minkowskian: in the distance metric, time h

This is a useful way to think about it, but you have to keep in mind that (even flat) spacetime is not Euclidean but Minkowskian: in the distance metric, time has an opposite sign to the spatial dimensions. So when you "rotate" a four-vector, it actually follows the surface of a hyperboloid rather than a sphere, which means that rotation has a discontinuity at Θ = π/4 (in normalized units) and the vector escapes to infinity! Only massless objects can travel "at the speed of π/4", everything else can only approach but never reach that speed.