His new movie is the suspense thriller "Secret Window," in which he plays a writer whose life disintegrates to a terrifying conclusion.
Depp is shooting his next movie in England, where CBS News Correspondent Mark Phillips sat down and talked with him.
There are few people in Hollywood who can get away with what Johnny Depp can. The actor has earned the right to walk around in silly hats after the spectacular success of his last movie, "Pirates of the Caribbean." But, he's in a state of showbiz denial.
"Movie stars are guys like Clark Gable, Humphrey Bogart, Robert Mitchum," says Depp. "Those are movie stars. I'm a guy with a strange job."
Depp has made a career out of being different. His Captain Jack Sparrow act in "Pirates" stayed away from the old "yo-ho-ho-and-a-bottle of rum" stereotype, and may have changed the way audiences will think of buccaneers forever.
And Depp's portrayal of tormented writer Mort Rainey in his new release -- the Steven King psycho-thriller "Secret Window" -- will also surprise audiences.
In the movie, Rainey goes through a hard divorce and some other changes. You'll have to see the film to find out more.
"For me as an actor, for an actor, this is a great, very challenging," Depp says of his character. "It's a genre I've never really investigated and its a very different kind of character … Its very rare to get a script that you have an emotional investment in by page six, and each page its like, 'Wow. Where is this thing going?'"
But then Depp has always had an attraction for the oddball character in mysterious movies. From what seemed like a mutant hairdresser in "Edward Scissorhands," to the ghost-ridden presence of "Sleepy Hollow," to "Pirates," Depp has tended toward the dark side and not to take himself or his work too seriously.
"Pretending for a living, you know, essentially telling lies for a living," Depp laughs. "It's a weird job, but I mean I've had weirder jobs. I sold ink pens over the telephone."
Now, Depp has been able to pick his roles and define them to his own liking.
"I've been in this racket for about 20 years now, and you know you go through phases where one week you're on the list, the A-list," explains Depp. "Then you're off the list for a month, or a year, and then you're back on because there's a movie coming out that might do well. And, then the movie flops and you're off the list. So, sometimes the road is bumpy. Sometimes the road is smooth. It's all the same to me. I just do my job."
And that road will take him wherever he wants to go. Currently, he's shooting another offbeat film, "The Libertine." The story is of an infamous freethinker and rake. Then it's a romp through a remake of "Willy Wonka's Chocolate Factory" and back to Captain Jack Sparrow and the "Pirates" sequel. All great fun while it lasts, and why shouldn't it?