I actually thought about this a few months ago. Used to be presidential elections elected the winner as president and the loser as VP. So you had the two most qualified participants from each side working together, with the one who commanded the most popularity among the citizenry/electoral college as the President.
I'd vote all day long for a Pres Obama VP Romney or vice versa ticket...ALL DAY LONG. We spend 4-8 years undoing each-other's fixes to social issues. Nothing would stun an electorate more than a qualified Presidential candidate picking a stonewall opponent from across the aisles who is just as qualified to be president...and then having the two work towards goals as leaders of their respective parties.