recently I started watching samurai champloo & someone asked mugen for his name.
he answered: ムゲンだけど
I was wondering what the role of だけど in this context was. I thought maybe it's the same as adding "..." after mugen in English.
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Your intuition is pretty accurate. It's expressing a feeling of "...what does that have to do with anything?" or "why are you asking?". Mugen doesn't understand why he's being asked his name.
A similar pattern would be:
A: How old are you?
B: I'm 30... (what does that have to do with anything?)
A: Whose car is this?
B: It's mine... (why are you asking?)