Of course there are those who feel there should be no relationship whatsoever. But the fact is you have an attractive male character going around with an attractive female character. It wasn't realistic back in Pertwee's day and it isn't realistic now to expect there to not be some sparks. (And in fact watch the final scene of The Green Death again; I defy anyone to deny that there wasn't something between the Doctor and Jo, and there are scenes between the Doctor and both Romanas where the sexual tension can be cut with a knife. Anyone who thinks it's all RTD's fault (or whoever wrote the TV movie) don't know the classic series at all, in my opinion.
It's not a situation that would be fixed by giving him a male companion, either, because even without the RTD influence there have been some speculation regarding Harry Sullivan, Turlough, Adric, Jackson Lake and even Wilf. Jack, of course, goes without saying.
Pretty much the only scenario some people would accept would be to bring back K-9. And even then, we'd have people griping whenever the Doctor made romantic sparks with a one-off character like Madame de Pompadour or Astrid Peth.
All that said it would be nice to see the student-teacher dynamic of Seven and Ace reestablished, and the twist of course being that we have a 1000-plus year-old Time Lord (906 my giddy aunt) who looks as young as his companion, yet being her teacher. It could add an interesting dynamic. Sort of what I'd love to see if they ever bring Jenny back full-time. (Which I suppose is another solution though some sick mind probably would come up with something...)
Something I doubt Moffat or his writers will do, but would be interesting, would be for Eleven to become frustrated because he ISN'T as irresistible to women as Ten was.
I also hope that all this talk about Karen being "mad as a box of cats" translates into the character. I'd love to see a "back away slowly from the insane companion" moment at some point!

Seriously, apparently Karen couldn't resist taking a bounce on a trampoline that for some reason was in the studio where they shot that SFX 3-D cover. I'd love to see the Doctor trying to teach Amy something and have to reel her in because she's gone off on a Luna Lovegood-style tangent!
Alex