Elixir of Life

The elixir of life, elixir of immortality or simply known as elixir (エリクシル) is a legendary/mythical potion, or drink, that is said to grant the drinker eternal life and/or eternal youth.

Principles
Alchemists are traditionally attributed to being able to create elixirs of immortality, though due to the fact that alchemists and magicians alike are searching for vampires in order to discover the creatures' secrets to immortality, it can be assumed that elixirs are either difficult to create, limited in effectiveness, or an impossibility.

Furthermore, elixirs are probably a means to an end for alchemists, who want to obtain immortality in order to achieve Ars Magna, the spell in which a alchemist could achieve their goal of simulation of the world inside the mind.

Index groups the elixir of life as one of the many ways to manipulate the dead, such as Vetala Sorcery, Necromancy, and the Hopping Zombie Spell. One can assume that the elixir of life not only gives immortality, but also gives life back to the dead.

Deep Blood Arc
Stiyl Magnus references that alchemists' main motivation is to create an elixir of immortality and turn lead into gold. In the Toaru Majutsu no Index anime adaptation, Stiyl asks Kamijou Touma what he knows about alchemists, to which he gives the standard answer of turning lead into gold and create an elixir of immortality.

Kazakiri Hyouka Arc
It is referenced after Index sees Kazakiri Hyouka's leg regenerate completely after kicking Sherry Cromwell's Golem down, and saving her.

World War III Arc
A Russian parallel of the Elixir the Water of Life is first mentioned in a story by Nikolai Tolstoy in his showdown with Vasilisa. A cannibal witch, is in charge of two fountains. One contains the Water of Life that gives one eternal life. The other contains the Water of Death that ends one’s life. The knights asked the witch to lead them to the Water of Life and they managed to monopolize the Water of Life by pushing the witch into the Water of Death. He manged to create it from unknown materials he found and sent for from the back-country of Russia sole purpose of killing Vasilisa. Unfortunately for him Vasilisa had the Water of Life inside her body that granted her youth and longevity while she denies acutely being immortal her self.