Diocletian died in retirement at his palace at Split in modern Croatia, and that's where archaeologists have just found what they believe to be part of his porphyry tomb - presumably they mean part of his sarcophagus, and are basing this on its shape.
This deep purply-red stone comes only from Egypt, and became very popular for Imperial burials under Constantine, but not so much before the empire became Christian. It was used for sculptures through the Imperial period, notably the set of Tetrarchs in Venice.
Later children of Byzantine emperors born whilst their father was on the throne were said to be born in the purple (Porphyrogenitos) because empresses gave birth in Constantinople in a room line in porphyry. The room is described by Anna Comnena, who was herself born in it in 1083 [Alexiad 7,2]:
... Nicephorus and Leo, the two sons of the Emperor Diogenes, who were born to him in the purple room after his elevation to the throne and were consequently styled " Porphyrogeniti." This purple room was a certain building in the palace shaped as a complete square from its base to the spring of the roof, which ended in a pyramid; it looked out upon the sea and the harbour where the stone oxen and lions stand. The floor of this room was paved with marbles and the walls were panelled with it but not with ordinary sorts nor even with the more expensive sorts which are fairly easy to procure, but with the marble which the earlier Emperors had carried away from Rome. And this marble is, roughly speaking, purple all over except for spots like white sand sprinkled over it. It is from this marble, I imagine, that our ancestors called the room " purple."
In Porphyrius' Passion of the Four Crowned Saints he describes Diocletian ('dilectatus in artem') hiring special carvers to sculpt capitals ('ex metallo porfiritico') out of porphyry ('ex monte porphyritico ... qui dicitue igneus') for him.
Diocletian - described as "infinitam quandam cupiditatem aedificandi" - built a fabulous palace at Split, the structure most associated with him today. Given the sources note his love of the purple stone, he can be seen as the genesis of it's extensive use by the emperors that succeeded him ... but it also means that unless we know more about the fragment found there, we can't assume it's a sarcophagus just based on the material.
Egypt fell to the Islamic armies of Mohammed's successors in AD 639, and the porphyry mines became inacessible to the Byzantines. Older sculptures continued to be re-cut and porphyry used as a material for small statues and heads but by then it had faded from fashion for Imperial inhumations - a number of Byzantine sources such as De Ceremoniis Aulæ Byzantinæ 2, 42 list where various emperors were buried, and the material their sarcophagi were made of, so we know the last porphyry sarcophagus used was for the joint burial of Marcian (died 457) and his wife Pulcheria (died 453) in the Church if the Holy Apostles in Constantinople.
The tombs of these early emperors, including Constantine, were looted by Alexius III Angelus circa 1200 to raise funds to bribe Henry VI Hohenstaufen not to invade. Constantinople was still sacked and looted as a result of the Fourth Crusade in 1204. The church itself was destroyed soon after the Ottoman conquest of 1453, to make way for a mosque of Mohammed the Conqueror - but not before the sarcophagi had been once again ransacked by men looking for loot but finding only old bones. The sarcophagi were scattered, many ending up in the Harem at the Topkapi, and are now in the ground of the archaeological museum. Although the De Ceremoniis Aulæ Byzantinæ described nine imperial porpyry sarcophagi in Constantinople these were not as elaborately decorated as those in Rome, so are now harder to identify - the easiest is the one described as cylindrical, as which belonged to Julian the Apostate (on the right in the photo above). The sarcophagus of Constantine the Great is described in sources as much bigger than the others, and so probably can be identified in the fragmentary sarcophagus decorated with Erotes gathering grapes (photo below).
Although no emperors are known to have used porphyry sarcophagi before Constantine, the so-called sarcophagus of Helena in the Vatican is decorated which scenes of was which suggest that it was that of her along with her husband Costantius I Chlorus. He was a contemporary of Diocletian, suggesting the practise could have gone back one generation.
I think I took these photos in May 2007, and because the sarcophagi are not labelled people tend to just walk past them - click on the photos to super-size them.