Yes, the direction of the time axis is as observer-dependent as is the rotation of spatial axes. In Cartesian geometry the distance (dx, dy, dz) between points depends on the choice of coordinate system, but the length of that 3-vector is always the same (compared to the distance between some reference pair of points)
Similarly in space-time it is only the length of the 4-vector between events A and B that can have any physical meaning. This length can be positive, negative, or zero. A positive length means the events are outside each other's light cones ("space-like"), thus have no causal influence on each other. A negative length ("time-like") means each could have participated in the events leading up to the other.
A zero-length ("light-like") means there is no event separation between the two, i.e. they are the SAME EVENT (this might be understood as length contraction in the direction of motion going to zero when v=c). All the points on A's light cone (which includes B) are the same event as A. All the points on B's light cone (which includes A) are the same event as B. So far there is nothing to determine the direction of time, and indeed the physics describing the transfer of action when v=c is symmetric with time.
But there are points Z on A's cone not on B's cone and these are also part of the same event AB even if they also have space- or time-like separation from B. Even if nothing can interrupt the ZAB event, the direct ZB distance could be time-like and thus set a direction of time, B before A. If B was a causal factor for A, it is difficult to see how ZAB could not be affected by the the direct BA interaction. It seems to me, here is the collapsing wavefunction. The only way I can reconcile it, is to introduce some time average long enough to allow all these paths to interact.