Someone who can't stop thinking and fretting about death suffers from thanatophobia. Beyond normal worries, true thanatophobia makes it hard to sleep, work, or enjoy life.
Thanatophobia has two Greek roots, thanatos, or "death," and phobos, "fear or terror." Sigmund Freud is credited with inventing this word, along with a theory that people who are terrified of death are actually dealing with unresolved childhood issues. Like a true phobia of heights, crowds, or snakes, thanatophobia causes intense anxiety to its sufferers, who may fear imagining their own death, or a loved one dying — or even the idea of talking about death and dying.