Why does Caesar's ghost appear in Act 4 of Julius Caesar?

Shakespeare employs ghosts and apparitions frequently in his writings, especially of characters murdered or wronged by their deaths. These ghosts generally have a twofold purpose: to haunt the guilty conscience of their murderers and avenge their deaths, and to act as an ill omen foreshadowing some oncoming doom. Caesar's ghost in act IV is remarkably similar to Banquo's ghost in Macbeth. Both are the ghosts of men murdered by a close friend, who appear to bring about their murderer's end.

Though Brutus became the de facto leader of the conspirators to kill Caesar, he was not the one who originally planned to kill Caesar. Cassius spurned him to action, suggesting that justice could only continue in Rome with Caesar's death. Brutus agreed and helped carry out the deed, defending it to himself and to the Roman people with tales of Caesar's ambition and impropriety as a leader. Yet the appearance of the ghost implies that Brutus's conscience may not be entirely clear. Caesar was his friend, and though he believes his actions were necessary, he feels some guilt for having murdered a man who trusted him so. When the ghost appears, Brutus asks it to identify itself:

Brutus. Art thou some god, some angel, or some devil,
That makest my blood cold and my hair to stare?
Speak to me what thou art.

Ghost. Thy evil spirit, Brutus.

The ghost's response could be interpreted as simply the evil spirit that is haunting Brutus. However, "thy evil spirit" can also be read as the evil spirit of Brutus himself. Caesar's ghost represents the evil in Brutus' heart that revealed itself when he murdered Caesar. This evil is now come to haunt Brutus.

Caesar's ghost also appears to impart a cryptic message to Brutus:

Brutus. Why comest thou?

Ghost. To tell thee thou shalt see me at Philippi.

Brutus. Well; then I shall see thee again?

Ghost. Ay, at Philippi.

Brutus. Why, I will see thee at Philippi, then.

This plants the idea in Brutus's head that doom awaits him at Philippi. When he arrives there and the battle has turned against him, he decides to end his own life, laying the blame for his death with Caesar's ghost:

O Julius Caesar, thou art mighty yet!
Thy spirit walks abroad and turns our swords
In our own proper entrails.

This fulfills the foreshadowing laid earlier in the play. When Antony discovers Caesar's body, he suggests that Caesar's spirit will have revenge, even in death:

And Caesar's spirit, ranging for revenge,
With Ate by his side come hot from hell,
Shall in these confines with a monarch's voice
Cry "Havoc," and let slip the dogs of war

Caesar's ghost has succeeded. His appearance has shaken Brutus's convictions, and the announcement that Philippi is the place where he and Brutus shall meet again drives Brutus to suicide at the end of the play as he accepts his fate as ordained by the spirits. Caesar's murderer has answered for his crimes.

In Julius Caesar, what is the significance of Caesar's ghost in Act 4, Scene 3?

At the end of act 4, scene 3, Caesar's ghost appears to Brutus.

It is time for bed, but Brutus is troubled. He asks Lucius to play the lute for him, but Lucius soon falls asleep. Brutus doesn't have the heart to wake him, so he turns to reading his book. However, in the light of his taper he sees a "monstrous apparition." Brutus says to it,

Art thou some god, some angel, or some devil
That makest my blood cold and my hair to stare?
Speak to me what thou art.

Thy evil spirit, Brutus.

What is the role of Caesar's ghost in Julius Caesar?

Many readers have questioned why Shakespeare's play should be called Julius Caesar when Caesar dies halfway through his play and the real tragic hero seems to be Brutus. Shakespeare was aware of that problem, and he seems to be trying to show in various ways that Caesar may have been dead but his spirit and indomitable will prevailed until the end. One of the ways Shakespeare reminds the audience of the spiritual presence of Julius Caesar after his death is by borrowing from Plutarch and dramatizing the scene in which Caesar's ghost appears to Brutus in his tent before the battle of Philippi.

BRUTUS:
Art thou any thing?
Art thou some god, some angel, or some devil
That makest my blood cold, and my hair to stare?
Speak to me what thou art.

GHOST:
Thy evil spirit, Brutus.

BRUTUS:
Why comest thou?

GHOST:
To tell thee thou shalt see me at Philippi.

BRUTUS:
Well, then I shall see thee again?

GHOST:
Ay, at Philippi.

BRUTUS:
Why, I will see thee at Philippi then.

[Exit Ghost.]                         IV.3

At the end of the play, both Brutus and Cassius state that Caesar was instrumental in their defeat at Philippi. When Brutus commits suicide by running on his sword in Act V, Scene 5, his dying words are:

Caesar, now be still;
I kill'd not thee with half so good a will.

Earlier, in Act V, Scene 3, when Cassius commits suicide by having Pindarus stab him, Cassius' last words are:

Caesar, thou art revenged,
Even with the sword that kill'd thee.

In that same scene Brutus comments:

O Julius Caesar, thou art mighty yet!
Thy spirit walks abroad, and turns our swords
In our own proper entrails.

All of this is to maintain the thesis that this play is about Julius Caesar and that he is still present in spirit even though he may have been assassinated in Act III, Scene 1, approximately the middle of the text. When Antony is speaking over Caesar's body in Act III, Scene 1, he predicts that Caesar will come back from hell.

And Caesar's spirit, ranging for revenge,
With Ate by his side come hot from hell,
Shall in these confines with a monarch's voice
Cry "havoc!" and let slip the dogs of war.

So the role of Caesar's ghost in the play is to demonstrate that Caesar is still very much present and influential in political and military affairs even though he may be dead. He is such a powerful person that even death cannot stop him from enforcing his will. He may not become a king himself, but his successor Octavius becomes an emperor and a god, and Caesar initiates a whole line of Roman rulers who use his name as a mark of highest honor.

From a practical standpoint, Shakespeare gets some extra work and additional exposure out of the actor who plays Julius Caesar. Shakespeare does something very similar with Banquo in Macbeth. Although the actor playing Banquo is seen being murdered in Act III, Scene 3, he reappears as a ghost at the coronation banquet in Act III, Scene 4 to horrify Macbeth and assert his presence as his murderer's nemesis. The actor playing Hamlet's father's ghost makes a reappearance in Queen Gertrude's bedroom in Act III, Scene 4 of that play. Shakespeare obviously liked using ghosts for stage effects.