Diana Gabaldon has said two interesting things about the iconic “Jamie’s ghost” scene right at the beginning of Book 1 (and 01×01 in the TV show):
1. Jamie is ~25 years old at that moment
2. The scene will be explained in the last scene of the last book
Jamie was about that age when he fought at Culloden. Readers know how that was one of the times he nearly died.
So I think that as he’s delirious after the battle, he somehow sees Claire – who he had sent back through the stones on the morning of Culloden – in her own time.
But he doesn’t remember any of this until the very final moments of his life – and we know how memories of Culloden are slowly coming back to him in Book 7 and Book 8.
But this memory brings him peace, because at the very moment he’s leaving earth – and, presumably, Claire – he’s remembering the moment right before she came to him.
Totally speculation on my part…but I think it could be plausible…