Does the radius of your cosmic horizon decrease as velocity increases? If I was moving at c-1 meters per second, and was observing an object moving 2 m/s away from my starting point as a result of the expansion of the universe rather than classical velocity, would I ever see the light coming from this object? Does that mean that as you move faster, the radius of your cosmic horizon decreases?
Your past and future lightcone surfaces are invariant: no matter what speed you have, at a certain point in space and time you can see the same things (although strongly Lorenz contracted and with aberration). So if you could see the object when at rest, you can see it near lightspeed.
The cosmic horizon is tricky since it is non-local: the distance where galaxies appear redshifted beyond detection changes with time because of the changing expansion rate of the universe. There are also visible galaxies that recede faster than light we can still see (!).
In some sense the horizon may seem to shift when you speed up because the redshift makes nearby parts of the universe undetectable, but this would be true even in a cosmology without any horizon. If you accelerate at a constant rate you can even get a true event horizon behind you where light cannot catch up to you (a Rindler wedge), but it does not have anything to do with the cosmology. Still, if the object is in that region it would be invisible. But that is because of acceleration, not speed.