Who Am I?

"Hi."
"Who are you?"
They both looked at each other. They were identical.
"I'm you," said one.
"No, I'm me," said the other. "Who are you?"
He thought for a second.
"If I'm not you, then who am I?"
"Maybe I'm you instead?"
A third identical person arrived.
"Who are you?" he asked.
The three of them looked at each other as they saw a group of other identical people approaching. It was like an endless hall of mirrors nightmare.