Question

what offer did the conspirators make to antony after Caesar's death?

Answer 1

Antony was given a position in the government. They also offered him to give away Cesar's funeral speech. They offered him a chance to give his assistance in appointing new officials of Rome.

Answer 2

When Mark Antony enters he asks the conspirators is if they plan to kill him, too. He also shows his devotion to Caesar by asserting that there would be no better place for him to die than by the side of Caesar and no better means for him to die than by "those your swords, made rich / With the most noble blood of all this world" (3.1.155-56).