Disclaimer: I don't own the Doctor, or any of the companions mentioned here, or, unfortunately, Doctor Who.
He carries around their dreams because it is easier than carrying around their memories, and he has always been a coward at heart.
And it figures then, doesn't it, that the other him, the reflection—
("No idea how you can be here, but there's only one person in the universe who hates me as much as you do.")
—well. The irony of the name is not lost on him.
And if he's quite honest with himself (which he normally goes to great lengths to never be), he has to admit that he feels much more like a Dream Lord than a Time Lord, these days. This body. There are cracks in the universe and silence lurks everywhere, and he feels like a naive, out-of-control schoolboy. He barely has a better grasp of what's going on than his companions.
("What if you were really old, and really kind and alone? Your whole race dead, no future. What couldn't you do then? If you were that old, and that kind, and the very last of your kind... you couldn't just stand there and watch children cry.")
He's become far too human.
Once—long ago and far away, when he'd been a different man—he'd pitied those who traveled with him, with their nightmares and bedhead and grogginess. All those wasted hours... But he'd learned. He can recall conversations over breakfast in which Nyssa described new worlds she'd created overnight; how she and Adric had dreamed competitively. The adorable way Sarah Jane had snored. Ace crying out in her sleep, as she'd never let herself in her waking hours.
Even now, after all these years, he can remember them all. Every single one.
It doesn't change a thing.
This is the truth that keeps him up at night; that follows him into the dark on those rare occasions he stops to rest. That his companions are only ever different to a point, and that they always end up more similar than they started out: ruined by the lack of him.
("Your friends never see you again once they've grown up. The old man prefers the company of the young, does he not?")
He dreams of them, and they haunt him.
Rose had dreamt of windswept beaches and shattered clockwork—that there were chipped-off pieces of her then-brittle heart just rattling around inside of her; that she was broken and everyone could hear it.
Martha had dreamt of unhealed wounds and stolen, shamefaced glances—that there was a hairline fracture hidden underneath her skin; that she was broken and no one would be able to tell.
Donna'd dreamt—Donna dreams—of fire and stars and a song that echoes across the universe—that there's somewhere, someone else she's supposed to be; that she's broken and she can't remember why.
He doesn't know what Jack dreams, or if he even can dream, living as he does now. It doesn't matter; that particular nightmare is a waking one. Jack doesn't have to dream that he's broken—he is.
And now he has two more to add to the collection. Rory's never-to-be-realized fantasy of a sleepy Leadworth future, and the compounding cracks in Amy Pond's increasingly fragile reality. (He wishes, more than anything, she could just wake up exactly where she belongs. He wishes he could wish her life into sense.) Two more lives he's ruined, just by touching them. More debts he cannot repay and mistakes he may never fix, despite how desperately he's trying.
("Save him. You save everyone; you always do. That's what you do."
"Not always. I'm sorry."
"Then what is the point of you?")
He is a nightmare and a broken promise and the worst person he knows.
("It said 'Satan.'"
"Come on, Rose. Keep it together."
"Is there no such thing? ... Doctor? ... Doctor. Tell me there's no such thing.")
We are each are own devil, and we make this world our hell.