In Julius Caesar, what do we know from Caesar's habit of referring to himself in third person?
Quite simply, it is Caesar's way of elevating himself above everybody else, god like almost. He is the only character in the play to speak of himself in the third person. In his mind, he is already emperor of Rome, and all that remains is the formal ceremony. That he refuses the crown three times is part of his strategy. It just makes the people want him more.
Cassius is correct in his observation of Caesar when he is trying to convince Brutus of the danger he poses to Rome. He compares him to a Colossus striding through Rome implying stepping on others along the way.
Are Cassius's fears correct? One has only to look at the history of Rome and the change from a Republic to an Empire to find the answer.
In my opinion, this shows that Caesar is at least a bit arrogant. Of course, it makes sense that Caesar should be arrogant given his position in their society.
The only people who refer to themselves in third person much these days are athletes who see themselves as larger than life. In Shakespeare's (and even today in extremely formal circumstances) you spoke to people in the third person if they were really important. As in "will the general please come this way, sir?" Or "will His Majesty please look at this."
So when Caesar does this, he's sort of giving himself credit for being important and that's arrogant.
Although I agree with both of these analysis', I also wonder why Cassius and Brutus often go back and forth between using third person and first. Why the waffling with Caesar, as well? He uses first person. Could it be, perhaps, to remind the audience of the play who is speaking?