"Everybody understands that Hillary's been one of the most important advisers that I have on a whole range of issues," Obama, describing Clinton as a "strong friend," said in the interview, recorded Friday at the White House.
Both Obama and Clinton deflected questions by CBS's Steve Kroft about whether the interview amounted to a presidential endorsement of the outgoing secretary state, who's at the centre of intense speculation about whether she'll make another bid for the White House in 2016.
Their appearance on "60 Minutes" has only further fired up the rumour mill.
"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, meantime, chided Kroft: "You guys in the press are incorrigible. I was literally inaugurated four days ago. And you're talking about elections four years from now."
Instead, the president said, he simply wanted to pay tribute to Clinton and the dedication she's brought to the job. He lavished her with praise throughout the interview, calling her an "extraordinary talent" and lauding "her discipline, her stamina, her thoughtfulness, her ability to project."
"I just wanted to publicly have a chance to say thank you," he said. "I think Hillary will go down as one of the finest secretaries of state we've ever had."
Clinton returned the kudos.
"I certainly am grateful for the president's steady hand and hard questions and thoughtful analysis on what we should and shouldn't do," she said.
She described their relationship as "very warm, close." Indeed, the pair chuckled warmly at times while listening to one another intently throughout the interview.
"I think there's a sense of understanding that sometimes doesn't even take words," Clinton said.
"We have similar views, similar experiences that I think provide a bond that may seem unlikely to some, but really has been at the core of our relationship over the past four years."
Such an astonishing love-in would have seemed unimaginable in 2008, when Obama and Clinton were locked in a brutal, bruising primary battle for the Democratic presidential nomination.
The "60 Minutes" interview aired on the fifth anniversary of the weekend Obama handily defeated Clinton in the South Carolina primary. That contest followed a particularly nasty leadup, when Bill Clinton angered African-Americans in the state when the former president, campaigning for his wife, described Obama's candidacy as a "fairy tale."
Many of the most incendiary rumours about Obama — including long disproven allegations that he's a Muslim and was born in Kenya — were propagated by Clinton supporters in 2008.
But what a difference four years —and a key cabinet position — has made between Obama and the woman who once derided him as being woefully unprepared for the White House.
Obama and Clinton were even photographed tearfully embracing and holding hands in September when the bodies of four Americans, including the U.S. ambassador to Libya, were returned to the United States following an attack on the consulate in Benghazi.
Clinton herself acknowledged that their interview Sunday was extraordinary considering the nature of their primary brawls. It's the first time Obama has been interviewed as president with anyone other than his wife.
"A few years ago it would have been seen as improbable because we had that very long, hard primary campaign," she said.
"And then President Obama asked me to be secretary of state and I said yes. And why did he ask me and why did I say yes? Because we both love our country."
Clinton added that if the roles had been reversed, and she had won the presidency in 2008, she would have "desperately" wanted Obama in a prominent position in her cabinet.
The "60 Minutes" interview took place after Clinton spent much of the past week on Capitol Hill, defending her handling of the Benghazi attacks. Ambassador Chris Stevens and three other American officials were killed by Islamic militants in Benghazi on the 11th anniversary of the Sept. 11, 2001 terrorist attacks.
Clinton forcefully denied anyone at the State Department or in the Obama administration made any efforts to mislead Americans on the attacks as the details emerged. Obama defended Clinton during the Sunday interview for her handling of the Benghazi crisis.
While Clinton's resignation marks the official end of her working relationship with Obama, many in the U.S. capital believe it signals the start of a new chapter in their association.
During last year's election campaign, Bill Clinton was one of Obama's top surrogates, despite his own notoriously frosty relationship with the commander-in-chief. Hillary Clinton suggested Sunday that her husband had a tougher time with her loss to Obama in 2008 than she did.
The buzz in the U.S. capital is that Obama promised to throw his support behind Hillary Clinton in 2016 in exchange for Bill Clinton's help on the campaign trail in 2012. The 41st president delivered a rousing speech about Obama at the Democratic National Convention in September.
If it's true Obama and the Clintons made such a deal, the president could find himself in a quandary if Joe Biden, his cherished vice-president, opts to throw his hat in the ring as well.
Biden's potential 2016 run for president did not come up in Sunday's interview, but the vice-president has hinted he intends to make a bid for the White House.
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