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"No hanky-panky in the TARDIS." Well, sure, this is family television after all. But in sharp contrast to the old series, since 2005 the relationships between the characters have been just as important to the show as the Monster of the Week. This has introduced a new subculture to Doctor Who fandom: "Shipping."
(Of note: There's a fantastic interchange between new series producer Phil Collinson and Eighth Doctor companion Daphne Ashbrook on the subject toward the end of Doctor Who: Podshock 141 [live at Gallifrey 20].)
In my opinion, the shipping idea has always been there; I see a lot of old and new fans who love couples like the Fifth Doctor and Nyssa or Tegan, the Fourth Doctor and Sarah Jane or Romana, Jamie and Zoe. Today's Internet had just made it easier for fans to connect and share their fanfic and artwork. And yes, RTD did make it a lot easier to talk about relationships in Who.
(And if you're going to write an alchemy story like you said, Rowling, they you pair off the hero and heroine! Re-read those classics you claim inspired you!)