Learn Turkish: Yeni komşu gelmiş mi? - Did the new neighbor supposedly arrive?

Geldi mi? uses the direct past tense, meaning you are simply asking "Did the neighbor arrive?" as a straightforward fact. You use gelmiş mi? (the inferential or reported past) when you are asking about something you wouldn't have witnessed yourself, often asking for gossip or hearsay. It translates closer to "Have you heard if the neighbor arrived?"

Why does the English translation say "supposedly arrive"? Is there a word for "supposedly" in the Turkish sentence?

There isn't a separate word for "supposedly". That meaning is built directly into the -miş suffix. Because this tense is used for things you didn't see firsthand, translating it with "supposedly" or "reportedly" helps capture that "heard through the grapevine" feeling in English.

Why is the question particle mi used here instead of mı or mu?

The question particle always follows 4-way vowel harmony based on the last vowel of the word right before it. The word before it is gelmiş, which has i as its last vowel. Therefore, the question particle must also take the i form, becoming mi.

If I wanted to ask "Did you supposedly arrive?", how would the end of the sentence change?