What role does the supernatural play in Julius Caesar?

From act 1 on, creepy sightings foretell a world out of joint. Casca, for example, runs into Cicero in act 1, scene 3, and describes to him some of the supernatural events he has recently witnessed. In one, a slave's hand was on fire as if lit by "twenty torches," yet it did not burn. Casca also passed a lion in the street, which walked by him without bothering him. In yet another instance, "a hundred ghastly women" were frightened because they saw men in flames.
Casca interprets all these supernatural happenings as omens of bad events coming to Rome. Cicero, however, says that while these events may be real, we can't presume to know what they mean:

Indeed, it is a strange-disposèd time.
But men may construe things after their fashion,
Clean from the purpose of the things themselves.


Of course, bad times are about to descend on Rome, and one of the themes in this play is the problem of how to interpret the signs from beyond. Julius Caesar, for example, should have paid attention to his wife's terrifying dream of violence, blood, and shrieking ghosts. She doesn't want Caesar to go to Senate the day following the dream, but he disregards her—and is assassinated.

Later, Brutus's fall is foretold through the supernatural appearance of Caesar's ghost in his tent.

In Shakespeare, killing the rightful leader of a country upsets the cosmos. In this play, supernatural occurrences are meant to be taken seriously as warnings of bad times to come. They add an eerie, otherworldly cast to what otherwise might simply be a play about politics and civil war.

Shakespeare knew his Elizabethan audiences and how to entertain them while also writing drama for the ages. He understood the power of the supernatural to engage the imagination, to thrill and intrigue through mysterious aberrations in the natural world that defy rational explanation. The characters and the conflicts in The Tragedy of Julius Caesar are those of the “real” world, but elements of the supernatural are found throughout the play, heightening the drama and contributing to the suspense.
The presence of the supernatural is introduced almost immediately in the play through the Soothsayer, who warns Caesar, “Beware the ides of March.” His certain knowledge of the future indicates powers that supersede those found in the realm of nature. The supernatural ability to foretell the future becomes a motif as the play develops. In Act II, the Soothsayer reappears, and augurers, interpreting a strange and ominous animal sacrifice, accurately predict Caesar’s death.
The play is also filled with terrifying, highly dramatic supernatural events. A vision of Caesar’s imminent murder comes to Calpurnia in a dream; the vision is fulfilled the following day as the conspirators bathe their hands in Caesar’s blood. The night before Caesar’s murder, Rome is assaulted with horrifying supernatural occurrences that defy nature. Calpurnia describes them to Caesar:
A lioness hath whelped in the streets.
And graves have yawned and yielded up their dead.
Fierce fiery warriors fight upon the clouds,
In ranks and squadrons and right form of war,
Which drizzled blood upon the Capitol.
The noise of battle hurtled in the air,
Horses did neigh and dying men did groan,
And ghosts did shriek and squeal about the streets.
Although Calpurnia has paid little attention to omens in the past, she is terrified by these displays of the supernatural. The ghosts that walk the streets of Rome prefigure a far more ominous spirit that appears in Act IV when the Ghost of Caesar, a “monstrous apparition,” materializes in Brutus’s tent and speaks to him of the future. “Thou shalt see me at Philippi,” the Ghost intones, predicting with certainty Brutus’s destruction. Like Calpurnia, Brutus is horrified to experience that which exists beyond the natural world.
Shakespeare’s plot and character development are more than sufficient to make the play compelling, and its universal themes and the poetry found within its lines elevate it from a period drama to an enduring work of literature. The Tragedy of Julius Caesar would certainly withstand the test of time without the inclusion of supernatural elements. With them, however, the play entertains and will continue to entertain on another level. Glimpsing a realm beyond the natural world will always fascinate, and it will never lose the power to amaze.