A NEW DYNAMIC DUO

At some point during his enforced banishment on Earth without access to the TARDIS, the Third Incarnation of the Doctor might have visited Gotham City in America.  There he could have teamed up with the caped crusader known as the Batman and they became friends and allies.  

Maybe.

In thanks for his help on a case (perhaps against a team-up of Mr. Freeze and the Ice Warriors?), Batman gifted the Doctor with a wardrobe taken from the Riddler during an earlier case.  This is why we saw the later incarnations of the Doctor wearing clothes with the question mark logo.  The gift would also include an umbrella with a question mark handle - a gift that was originally given to the Riddler by the Penguin.....

BCnU!