In the current cosmological model the observable universe is limited by a cosmic horizon. We can't see galaxies beyond the horizon. Imagine a spaceship in the vicinity of our horizon. If the ship is stationary relative to the local galaxies there, we would see it deeply redshifted. Now the ship takes off in the direction away from us with a relativistic speed relative to the local galaxies. Would the ship disappear for us even though it still is not as far away from us as the cosmic horizon? Or would we still see the ship just even more redshifted until it crosses the cosmic horizon?
By definition, the edge of our observable universe is the most distant location from which we can obtain information. Since nothing can travel faster than a beam of light, you can also think of it as the location of a hypothetical photon that has been travelling away from us since the Big Bang (I say hypothetical because the early universe was opaque to light). This means that our observable universe continues to increase, as light keeps travelling further away. Locally, near the position of the spaceship, the edge of our observable universe is receding at the speed of light, so the spaceship can never cross it. The spaceship will always remain observable. See also this post for additional information.
As to what happens to its redshift, that's a more complicated question. It will depend on the velocity of the spaceship. The redshift of co-moving galaxies near the edge of the observable universe is decreasing (see this post for the calculations). If the spaceship is fast enough, it could overcome this effect and its redshift could instead increase.
I think that the answer is your last statement I dont understand why you think it would disappear. The event horizon is like the surface of a sphere with the observer at the centre. The observer has an obervable universe within that sphere. When an object pases out of that sphere it is not observable any more.