(PRATTO) KATA TON DAIMONA EAUTOU: This ancient Greek phrase is written on the grave of James Douglas Morrison, known to the entire world as Jim Morrison, lead singer of DOORS. Morrison in July 3rd, 1971 was found dead in the bathtub of the Parisian apartment, where he stayed with his girlfriend, Pamela Courson. The official cause of death was heart attack, though no autopsy was made to determine the exact causes. Morrison was buried in the famous Parisian cemetery, Père Lachaise, the larger and most notable cemetery, as in there are buried Edith Piaf, Frederic Chopin, Marcel Proust, Oscar Wilde and many more famous (and not).
The phrase, written on a bornze plaque, set on his grave, was selected by Morrison's father, George Stephen Morrison in 1990.
(source: https://danielstreett.com/2012/10/31/greek-rock-stars-and-tombstones/)
When we read the inscription, our mind might wander to demons and, if we combine it with the troubled life of Morrison, then we might believe that the true meaning of the phrase is that Morrison couldn't fight with his demons.
But, the word "daemon" (δαίμων) in ancient Greek has little to do with demons, the devil, Lucifer etc. In Greek theology there were no demons, only gods and that is the meaning of the word DAIMON. As we can find in Lidell Scott lexicon, daemon is: god, goddess, of individual gods or goddesses, and also the power controlling the destiny of individuals: hence, one's lot or forlune.
http://www.perseus.tufts.edu/hopper/morph?l=dai%2Fmwn&la=greek&can=dai%2Fmwn0
In other words, DAEMON (δαίμων) was the god/goddess that controlled one's person destiny, and that why when a person was happy, blissful in ancient Greek was called EYDAEMON (ευδαίμων). And more over, "DAIMONA EAUTOU" was called the deity that lived inside every human, his whole life, from birth to death. This deity was the one that cared for the prosperity, something like a christian Guardian Angel.
There isn't any exact reference of this phrase in Ancient Greek texts. So, if we had to give an exact translation, it would be: (ACTING) ACCORDING TO HIS OWN DAEMON.
In true essence, though, in my opinion, the epigraph means : ACTING ACCORDING WHAT MY CONSCIENCE THINKS IS RIGHT, a way of life approved by the Stoics philosophers, who declared that the main aim in a person's life is to live according to his nature, using his logic and that will lead him to Virtue.
Morrison tried to enter the dark realm of human's unconscious and hidden desires. He surely did live his own way, following his own rules.
Thanks for reading!