The supernatural events in Julius Caesar are very much part of ancient literature. The Romans were very aware of supernatural events such as prodigies and omens. They believed that the gods communicated with them through these. We can say that they were like signaling mechanisms with the divine. This might seem strange to modern readers, but it is only in the west and only recently that people shied away from the supernatural. Even in Shakespeare's time, people had very strong beliefs in the supernatural. So, I would say that these supernatural events fit right in the narrative and also fit into the culture of readers with perhaps the exception of modern times.
To begin with, the supernatural elements in Shakespeare's "Julius Caesar" are very significant because they impress upon the contemporary Elizabethan audience the importance of the Divine Right of Kings. According to this theory of royal absolutism the King was the representative of God himself on earth. The same principle is evident in all of Shakespeare's history plays and "Macbeth." Similarly, since Julius Caesar was the Roman Emperor he was also God's representative on earth and any treasonous act against him is foreshadowed and accompanied by bizarre supernatural happenings.
Secondly, Shakespeare has included supernatural elements in "Julius Caesar" to create an awesome effect in the minds of his contemporary audience by taking advantage of their superstitious beliefs in the supernatural. Some of the important examples of the supernatural in the play are:
In Act I Sc.3 Casca and Cicero meet on a Roman street in the evening. The weather is terrible and a storm is raging and both the heaven and the earth seem to have been shaken by the Gods above leading Casca to remark:
Cicero immediately asks him what other terrible sights he has seen and Casca lists out for him all the weird things seen by him. Two of the most striking 'supernatural' events described by Casca are (1) the slave who was completely insensible to his hand blazing away like twenty torches burning together and not being scorched at all and (2) the nocturnal owl hooting and shrieking in the market place at noon.
Calpurnia, Caesar's wife appears only once in the play in Act II Sc.2. She is presented as a very troubled and anxious lady deeply concerned about the safety of her husband. She pleads with Caesar not to go to the Senate because there have been reports of very bizarre happenings in Rome and she herself has had a terrible dream. Just then Decius arrives to accompany him to the Senate, and Caesar narrates to him Calpurnia's dream and tells him that he won't be coming to the senate:
"Calpurnia here, my wife, stays me at home;
She dreamt tonight she saw my statue,
Which like a fountain with an hundred spouts,
Did run pure blood, and many lusty Romans
Came smiling and did bathe their hands in it.
And these does she apply for warnings and portents
And evils imminent, and on her knee
Hath begg'd that I will stay at home today."
Calpurnia foresaw in her dream the assasination of Julius Caesar. She saw in her dream the statue of Julius Caesar being transformed into a fountain from which spouted not water, but blood and the Roman citizens smilingly washing their hands in his blood.