Both Obama and Clinton batted away questions about future campaigns, but the joint interview - the president's first with anyone other than first lady Michelle Obama - was only likely to increase the fascination with Clinton's future.
"The president and I care deeply about what's going to happen for our country in the future," Clinton said. "And I don't think, you know, either he or I can make predictions about what's going to happen tomorrow or the next year."
Obama, who suggested the joint interview as Clinton prepared her exit from the State Department, lavished praise on his rival for the 2008 Democratic presidential nomination. He called her a friend and an extraordinary talent, and praised "her discipline, her stamina, her thoughtfulness, her ability to project."
It teetered on an endorsement of a 2016 presidential bid that is still an open question. Clinton advisers say she has not made a decision about a run, while Democratic officials suggest Clinton would be an early favorite if she decided to mount another campaign.
Obama and Clinton laughed when asked about the political future.
"You guys in the press are incorrigible," Obama said when pressed on another Clinton presidency. "I was literally inaugurated four days ago. And you're talking about elections four years from now."
The possibility of a presidential campaign for Vice President Biden did not come up during the interview, taped Friday at the White House.
Obama described why he insisted Clinton become his secretary of state.
"She also was already a world figure," Obama said. "To have somebody who could serve as that effective ambassador in her own right without having to earn her stripes, so to speak, on the international stage, I thought, would be hugely important."
It was a job she initially refused. But Obama kept pushing, Clinton said.
"The one thing he did mention was he basically said: 'You know, we've got this major economic crisis that may push us into a depression. I'm not going to be able to do a lot to satisfy the built-up expectations for our role around the world. So you're going to have to get out there and, you know, really represent us while I deal with, you know, the economic catastrophe I inherited."