I was as happy as many others in seeing Obama ask Republicans to let cameras in to his Q and A period with them, and to see the Republicans agree to it. I've written in support of the idea of an American version of Question Time before, and now there's a bipartisan/nonpartisan group organizing to make it happen (sign the petition!). I think there's a real chance for it - usually it's the opposition party that wants to question the president, but now we have a president who's pretty confident, so maybe both will go for it.
Nate Silver posted his own ideas on how it could work. My opinion is there should be two types of question time. One type emphasizes accountability - this would be a more formal process, more like what parliamentary systems do and what Obama did with the Repubs. The other would emphasize cooperation and policy development under potential bipartisan conditions - this wouldn't even be done at Congress, but with the president sitting down in informal but televised situations to talk to one or two Republicans to give them a chance to convince him that he should adopt a policy of theirs.
The question time with the Republicans was formal, but emphasized bipartisanship and cooperation. I don't think they need to combine it that way. There's nothing wrong with the Republicans trying (and failing) to critique Obama, and other circumstances would give better situations to work on cooperative ideas.