William Shakespeare loved to include elements of the supernatural in his plays. First of all, the audience sincerely believed in witches, ghosts, demons, etc. Interestingly, it was Shakespeare that introduced fairies as playful mischievous creatures; before, they had been presented as malevolent beings, much like witches and demons. Elizabethan audiences believed that demons and witches worked for the devil in order to trick souls to their everlasting damnation. They also believed that ghosts could not make a human being do something, but since ghosts could not do anything themselves, they would try to get a human to do their work. This might even mean uncovering hidden treasure.
When Elizabeth I died, Shakespeare continued to include the supernatural, this time because Elizabeth I's heir to the throne, her cousin's son James, was fanatical about the supernatural, even writing a book called Demonology. (James I also persecuted alleged witches.)
It is, therefore, not a surprise that Shakespeare's Julius Caesar has elements of the supernatural, as did Hamlet and Macbeth, to name a few.
Brutus, one of the conspirators who murders Caesar, sees a ghost. (It is from here that the old saying, "Great Caesar's ghost" probably comes.) Brutus sees the ghost twice, once at his tent and once on the battlefield. The appearance of the ghost is very disturbing to Brutus.
Late that night, Brutus is visited by Caesar's ghost, who states that he will see Brutus at Philippi.
A soothsayer (fortune teller, prophet, oracle) warns Caesar twice to beware of "the ides of March" (March 15, probably the day of the full moon, another omen). Caesar doesn't listen to the soothsayer, ignores the advice, and is ultimately assassinated.
A third example comes in the form of Calpurnia's dream in which a statue of Caesar is spouting gouts of blood (foreshadowing of Caesar's assassination).
When Decius Brutus, one of the conspirators, comes to escort Caesar to the senate house, Caesar says he will not go, confiding that Calphurnia dreamed she saw Caesar's statue spouting blood.
Decius Brutus allays Caesar's concerns, insisting the dream is a sign of good luck. At first Caesar is concerned about the dream, but then he is ashamed that he almost gave in to Calphurnia's fears, and he decides to go out, against his wife's advice.
The presence of the supernatural in Shakespeare's plays is not at all unusual. An article I once read stated that it may be because of Shakespeare that Tinkerbell is such a sweet fairy; before Shakespeare's time, only dark things came from the world of the supernatural.