People say the Master is the Doctor’s Moriarty, and they’ve been saying that since the character was introduced in the 70’s, but I say that’s not quite accurate.
He may be the Doctor’s nemesis, but before that, he used to be his Watson, and a good chunk of classic episodes still have them working together like that. But we know that already. What’s interesting is that he’s the Doctor’s Irene Adler, too:
Trying to outwit the Doctor through clever disguises and keeping a step ahead of him, thereby always managing to escape (and with style, to boot), I mean. And the Doctor being impressed/charmed enough by him that he calls the Master his Best Enemy.
…well, and the bondage stuff and violation of personal space, if you prefer BBC’s Sherlock. To say nothing of the on-again-off-again canonicity of their ships (same goes for the Watson aspect).
I just find it neat that you could argue he’s all three of those people to the Doctor, because those are three of the most popular characters in the Holmes stories.