> If you kill your grandfather in the past you modify the timeline and therefore it's another timeline, by definition.

that is correct if you go by the M-theory, Many world interpretation of quantum physics.. or the multiverse theory.
If you kill your grandfather and can speak about it, you didn't kill your grandfather, beacuse you exist in the universe where you didn't kill him, beacuse you exist..
Almost like quantum immortality.

experiment: Quantum suicide.
A physicist sits in front of a gun which is triggered or not triggered depending on the decay of some radioactive atom. With each run of the experiment there is a 50-50 chance that the gun will be triggered and the physicist will die. If the Copenhagen interpretation is correct, then the gun will eventually be triggered and the physicist will die. If the many-worlds interpretation is correct then at each run of the experiment the physicist will be split into a world in which he lives and one in which he dies. In the worlds where the physicist dies, he will cease to exist. However, from the point of view of the non-dead physicist, the experiment will continue running without his ceasing to exist, because at each branch, he will only be able to observe the result in the world in which he survives, and if many-worlds is correct, the physicist will notice that he never seems to die.