Definition of immortal human is blurry. Is that also include that you cannot suicide? So, even if you do not age you are actually trapped in time and space.

What would happen if i try yo vertically tear myself while i am immortal?

Respawning sounds more interesting rahter than being immortal. They store your memories in some way(like some implant on your body sends everything you live,see,hear,feel back to a server) and when your body dies, they create you a new body(clone) that looks exactly like your previous body and load your memories to that body? Would this person still be you? This is more like immortality with possibility to die.