It'll make you feel like Cormac McCarthy.
This is a pretty common writing prompt. You might have even done it before. If so, or if you just want to make it a bit more challenging, try this: introduce a third person. Using only dialogue, write a conversation among three people in such a way that the reader knows exactly who said which lines.
Ulterior motive
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You have a number of tools to use to create a character's voice: word choice, the use of space fillers, speech impediments, consistent speech errors (think Mrs. Malaprop), and even accents (though I should warn you against overusing eye dialect). So give it a shot. And have some fun with it.