WMG: Indiana Jones and the Temple of Doom
WMGs for Indiana Jones and the Temple of Doom. Warning: Potential unmarked spoilers.
Mola Ram is both immortal and is Sankara, the original possessor of the Sankara Stones.
Indy makes specific reference to this character early in the movie, when he shows a parchment to Short Round that explains the story of Sankara, a hero who was given five magic stones from the goddess Shiva in order to go combat evil. The stones were supposedly lost on his death.
Indy's invocation of Shiva at the end of the movie is very specific: he accuses Mola Ram of betraying Shiva, over and over. But how can an accusation like that stick if Mola Ram is a Thuggee priest of Kali? How exactly has he 'betrayed' Shiva at any point? Answer: he was the original hero who possessed the Stones, but he turned away from the goal of helping fight evil in order to serve Kali, and use the Stones to achieve world domination. While struggling for the stones on the rope bridge, Mola Ram yells very empatically: "The Stones are MINE!" - and this is what clues Indy in to Mola Ram's real identity.
How did he lose the stones to begin with? It's canon that Mola Ram's body was never found. It's possible that he was eaten alive by alligators completely, but perhaps he can survive death - and his first 'death' was caused by loyal Shiva-followers, after his first conversion to Kali. This was when the stones were taken from him. He'd spent the rest of his immortal life trying to find them again, and founded the original Thuggee cult in the meantime.
We see more immortal beings in the series: Indiana Jones and the Last Crusade
shows us a knight who had been alive since the Crusades. Given that the Sankara Stones are also holy relics, perhaps they can give immortality too.