There are not, however, any _timelike_ unit vectors
I hate to nitpick my own note, but here goes:
Minkowski space is (1,3)R ... that is, we have a
metric with signature -+++ and we have real scalars.
There also exists (0,4)C ... that is, metric
signature ++++ and complex scalars.
In this latter space it is trivial to construct
timelike unit vectors: If u is a unitoid timelike
vector, then just multiply by i. That is, (i u) is
a unit vector.
Since (1,3)R is a subspace of (0,4)C, there is no
doubt that you could do SR in (0,4)C if you wanted
to. Indeed it was formerly fashionable to do so.
But there is no real advantage to doing so, and
multiple disadvantages. The ++++ metric doesn't
change the physics: timelike intervals are negative
while spacelike intervals are positive. (0,4)C makes
the metric look nicer at first glance, but it doesn't
really simplify the calculations; calculating
sin(i rho) is not any easier than calculating
sinh(rho).
Spacetime is not Euclidean. The mathematics of (1,3)R
is the "tightest fit" we know for the structure of
spacetime, while (0,4)C is less tight.
For a vehement deprecation of imaginary time coordinates,
see Misner/Thorne/Wheeler _Gravitation_ page 51 (box 2.1,
"Farewell to ict").