According to IGN Portugal and translated by NintendoLife, The Legend of Zelda: Breath of the Wild has an alternate ending. This was confirmed by the game's producer Eiji Aonuma. Unfortunately he didn't give any other details, but said that some players will experience a slightly different ending.
"There is a certain criteria you have to fulfil to see it. Essentially, you do some specific things and you'll see a different ending."
Alternate endings are nothing new in Zelda games, but the differences between different endings have always been very subtle. Breath of the Wild takes the franchise in a new direction, so maybe the alternate ending is something to look forward to.