By definition, death is the end of life. Anyone who wants to argue otherwise has a pretty tough argument to make.
While we don't 100% understand the human brain, we do know enough to say that that's where consciousness happens. Drugs (including hormones), tumors, and physical damage to the brain affect the person's personality. Monitoring the amount of electrical activity in a brain can tell the difference between someone awake, in deep sleep, or dreaming.
In the Middle Ages it may have made sense to think of consciousness as a fifth element, not reducible into smaller elements. But now that we do know what a brain is made of, it's clear that consciousness requires an extremely complicated physical structure.
Stories of "ghosts" were invented as then-simple-seeming explanations for mysterious phenomena. A lot of those phenomena have since been explained (if you're ever dared to stay in a haunted house, bring a carbon monoxide detector).