The original English definition of the German loanword doppelgänger, which translates literally to double goer, is a spirit double or counterpart of a person. The concept has a long history in European folklore. Although that definition remains, the word is now more commonly used to denote someone who looks exactly like someone else, especially when the two people don’t know each other.
Although there is nothing wrong with using the umlaut over the a in doppelgänger, English is not kind to umlauts, and the mark is omitted as often as not. The word has been in English for about two centuries, so it now goes unitalicized and without quotation marks, and there’s no need to define it on each use.
Examples
After Nick, on the run, ponders the improbability of having stumbled across a doppelgänger whose identity he could easily steal, he learns the doppelgänger may have been struck by lightning. [New York Times]
There are few unlikelier places to have encountered what may have been a doppelganger than the wilder reaches of Wyoming. [Haunted Heritage, Michael Norman and Beth Scott]
A Barack Obama doppelganger who was cut short and ushered off stage at a Republican conference moments before turning his attention to presidential hopeful Michele Bachmann has received death threats. [Independent]
I think all white people have a black doppelgänger and vice versa. My dad’s black doppelgänger is the father in The Fresh Prince of Bel-Air. [Your Voice in My Head, Emma Forrest]
Share pictures of yourself or family and friends who could be doppelgangers of characters in the movies (extra points to those who dress up in costume). [NOLA.com]