Pink Elekk

A Pink Elekk is a pink colored elekk that is a hallucination. Pink elekks are only visible while the player character is drunk from alcoholic beverages.

Terokkar Forest
Pink elekks are found around Allerian Stronghold and Stonebreaker Hold.

Brewfest
Pink Elekks are connected with the Brewfest event and were a quest objective of and  during Brewfest 2007. The quests are in hiatus for Brewfest 2008.

Per these two quest, the pink elekks were visible in the beer garden celebrations outside of most of the major cities: "Visit the beer gardens outside of Stormwind, Darnassus, and the Exodar, zap three elekks at each location" (Alliance) and "Visit the beer gardens outside of Silvermoon, Thunder Bluff, and the Undercity, zap three elekks at each location" (Horde). However, though the quests to "disperse" them have been removed, the pink elekks were not removed and are still visible in their Brewfest 2007 locations just as the Wild Wolpertingers are still around, although the quests to catch them are also in hiatus.

Per the quests, they are "dispersed" with an and cannot be engaged normally (otherwise you would not need to use the quest item to disperse them).

Inspiration
Seeing pink elekks when you are drunk is a reference to the expression, '', a euphemism for for drunken hallucination. The expression was an inspiration for the musical segment in the Disney movie  where in the main character accidentally gets drunk and hallucinates about pink elephants; you might assume that many players will catch the connection to  rather than.