Imagine what takes place 20 years after The Odyssey ends.

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Another poet has already answered this question in the form of the poem "Ulysses," where Lord Alfred Tennyson imagines what Odysseus (Ulysses is another name for him) would be like after he arrives back at Ithaca and defeats the suitors. It would be an interesting exercise to base the response to this question around this poem. Its basic argument is that a character such as Odysseus would get incredibly bored with "normal" life that does not involve danger, excitement and violence. Having spent so many years of his life chasing adventure and glory, the rather staid duties of king would be very difficult for him to settle for. The reader can detect hints of this obsession with glory in Book 11 when Odysseus has a conversation with the dead Achilles in the underworld. Note what he says:

But you, Achilles,
there’s not a man in the world more blest than you—
there never has been, never will be one.
Time was, when you were alive, we Argives
honored you as a god, and now down here, I see,
you lord it over the dead in all your power.
So grieve no more at dying, great Achilles.

Here, Odysseus tries to reason with Achilles and argue that actually he is better off dead than he is alive, because he has gained glory that will never fade and he is able to be worshipped as a god. This is something that could be used to indicate that Odysseus is first and foremost an adventure seeker who would find it impossible to settle for a quiet life. Twenty years after the end of this text therefore would see Odysseus thoroughly bored and craving adventure.