Does Winston die at the end of 1984, or is the image of the bullet in his brain at the Ministry of Love a dream?
As one of the previous answers pointed out, Winston has been changed from the rebellious individual into a completely obedient and subservient member of the party. This change in some ways mimics Winston's job as a member of the Ministry of Truth. His task there was to find and destroy any reference to events or people in the past that the Party wished to erase. Once found, they were changed and edited out of existence and any trace of them was dropped down the "memory hole."
The bits of Winston that the Party wished to erase have been erased. He no longer feels any anger at the party. He no longer feels the lust and desire he had for Julia. He no longer feels a connection to O'Brien whom he thought was a co-conspirator. He cannot even wish for the reality of a bullet ending his life as it has now been given wholly to the Party.
Had they "allowed" him to die, the Party would have lost. By keeping him alive but reducing him to his totally servile state, they have triumphed.
No, Winston is still alive. When the book concludes, Winston is waiting for the Party to end his physical existence. Winston has been re-educated, cleansed, and reborn with O'Brien's help in the Ministry of Love, so he is now ready to die.
Julia is in the same predicament.
Death is a desirable gift that the Party offers to all "healed" thought criminals.
No. He dies spiritually and psychologically at the book's end. His independent spirit is completely crushed, but he is left alive to love Big Brother. However, that image to which you refer is an indicator of how completely dead he is in these other ways: they might as well just shoot him. He's no longer any threat to them because he's dead inside.
In George Orwell's 1984, Winston does not physically die at the end of the book. He dies figuratively, however, at the end of 1984.
During the story, Winton lost his individuality to the Ministry of Love, all the unique characteristics that made Winton be himself and comprised his personality have disappeared. And he has become the re-trained puppet of Party. In essence, Winton's personality and soul have be shot by O'Brien, but physical Winton is still there; only a shell of his previous self.
While sitting next to the chess board in the Chestnut Tree Café drinking gin, Winton remembers a happy childhood memory which could symbolize his life flashing before his eyes.
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.--quote sited from ebooks.adelaide.edu
Here Winton basically repents and purges himself of past actions, in preparation for death 'soul white as snow'. Subsequently, the image of the bullet in his brain represents Winton spiritually dying and the physical Winton who remains is the re-trained puppet of Party.
He desires to be dead as a gift from the Party for his disloyalty previously throughout the book, which will show his "devotion and loyalty" to the Party.