I think this conflict was included because Shakespeare was trying to show that all was not perfect for either the conspirators or for those who determined to fight against them. As much as Brutus wanted to believe that the assassination of Caesar was done to restore the glory of Rome, there really wasn't much glory in Rome to be restored, nor had there ever really been. The conspirators claimed that they wanted to prevent the horrors that would occur if Caesar were crowned emperor...and yet, Rome has been plunged into civil war because they went that route and murdered Caesar!
In addition, the conflict between Brutus and Cassius stands in contrast to the conflict between Antony and Octavius, who are fighting against the conspirators, but can't seem to agree on much of anything, despite the fact that they are now part of the triumvirate that is supposedly ruling Rome.
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