Yes, but that kind of motion (tachyonic) is unphysical. It violates causality: you can imagine that by "rotating" the "horizontal" motion very slightly "downwar

Yes, but that kind of motion (tachyonic) is unphysical. It violates causality: you can imagine that by "rotating" the "horizontal" motion very slightly "downwards", in a different reference frame it is backwards time travel. The salient thing is that there are three disconnected classes of 4-vectors: timelike (massive particles), spacelike (tachyons), and lightlike (massless particles). Lorentz coordinate transforms can only take you between 4-vectors of a particular type: e.g. there's no inertial reference frame where timelike motion looks like tachyonic motion.