What happened to Martin Septim after the Avatar of Akatosh became stone? (Spoilers)

it's implied that the Avatar of Akatosh is Martin

Once the final cutscene is over, Ocato will run up to you and ask about Martin: "What happened? Where's Martin? I must congratulate him! Mehrunes Dagon is defeated! Cast back into Oblivion! We've won!" When you tell him Martin is gone, he says: "What do you mean, gone? We saw the Temple dome explode, the avatar of Akatosh appear... that was Martin?"

Source: The Unofficial Elder Scrolls Pages > Oblivion:Brother Martin > Quest-Related Events > Aftermath

the reason Martin was able to became Akatosh's Avatar is because his bloodline descends from Saint Alessia

The Dragon God imbued Alessia's line with blood from his own heart, initiating a sacred compact which kept the Dragonfires lit and Oblivion's influence barred from Mundus, so long as those of the Dragon Blood wore the Amulet of Kings. Ever since, the legitimate rulers of the Empire have been called Dragonborn: Alessia's successors, Reman Cyrodiil and his heirs, and finally the Septim Dynasty. Indeed, the title is particularly associated with Tiber Septim himself, perhaps because he made notable use of the thu'um in his early conquests. Only these true rulers, approved by Akatosh, could wear the Amulet of Kings and light the Dragonfires.

Source: The Unofficial Elder Scrolls Pages > Lore:Dragonborn > Dragonborn Emperors

one of the qualifications for a Septim Emperor is that they use the Amulet of Kings to relight the Dragonfires. to do this the Emperor-to-be (Martin) has to have Akatosh's blood in them.

given that people say Martin died we can assume the giant Akatosh Statue that's coming out of the Temple of the One has no life in it so Martin is dead