I will pick the conflict of Octavius vs. Mark Antony in act 4, scene 1.
In this scene, Octavius and Mark Antony argue about Lepidus. Antony believes that Lepidus is extremely incompetent, and he cannot believe that he is going to be the one to share the government with them. He is very mean and critical of Lepidus. On the other hand, Octavius is more positive and he tries to defend Lepidus by arguing that he is a good and brave soldier.
This conflict helps develop the plot by showing the difficulties that the two will have to face in order to maintain power. They will have to kill many people, including people they truly value. They will also have to learn how to share power. They will have to learn how to respect each other and work together. Finally, they will have to find a way to face the armies of Brutus and Cassius.