In 1984, does Winston die from a bullet at the end of the book or is he in a dream-state?

Winston survives all the way to the end of George Orwell's 1984. The end of the story finds Winston at the Chestnut Tree Café, sitting by a chess board and drinking gin. A number of memories appear in his head. At first he remembers a day from his childhood, before his mother disappeared. It was a memory of joy, and Winston "pushed the picture out of his mind. It was a false memory. He was troubled by false memories occasionally."

Of course, this was a real memory, but he has now been "trained" to believe only those thoughts that align with the Party's goals and pronouncements, which means forgetting all that made Winston unique.

After a bulletin announces a grand victory in Africa, Winston silently rejoices in the victory of the Party and soon slips back into a "blissful dream":

He is back in the Ministry of Love, with everything forgiven, his soul white as snow. He was in the public dock, confessing everything, implicating everybody. He was walking down the white-tiled corridor, with the feeling of walking in sunlight, and an armed guard at his back. The long-hoped-for bullet was entering his brain.

Of course, no one at the Ministry of Love murdered Winston, even though O'Brien threatened (or promised?) that Winston would eventually be shot. But O'Brien and the Ministry of Love did murder Winston's self. At the end of the novel, Winston no longer exists as a thinking individual. He exists only as a puppet of the Party, forever selfless, forever loving Big Brother.

Winston's self is the part that makes him human and unique — it essentially is Winston. And now that it is dead, he waits only for his soulless shell of a body to die as well.