My friend: I was there together with (Jon). I wanted to speak with him.
I didn't hear well the name, or the name wasn't clear to me, and I asked him:
Me: With whom?
It that correct to ask this way or should I ask in full sentence such as:
With whom was you there? or Did you want to speak with whom?
"With whom" is grammatically correct, a little formal but that is acceptable. You could improve your phrasing with some of the social language:
My friend: I was there together with (Jon). I wanted to speak with him.
Me: Sorry, I didn't catch the name. With whom did you want to speak?
I've added an apology (I'm British, we say "sorry" a lot) and I've cued the question with an explanation ("I did hear", not "I heard but I don't believe it"). It would be equally correct to ask "Who did you want to speak with?" A full sentence here is more friendly.
I believe you're right, following this tip: https://www.grammarly.com/blog/who-vs-whom-its-not-as-complicated-as-you-might-think/ You can replace Jon with "him" but not with "he". So, to me, it's totally correct to ask "With whom?".
But if your friend says :
Your friend: Jon was there together with me.
Then you reply:
You: Who was there together with you?
But allow me to correct the two questions you might also ask. I think it's better to say :
With whom were you together? or You were together with whom?
(I don't think it's necessary to repeat "there".) And,
With whom did you want to speak? or You wanted to speak with whom?
"With whom?" is fine. We might note that many native speakers will say "With who?", which is wrong but very common.
An abrupt question like that might come across as rude, depending on how well you know the person, how formal this conversation is, etc. So it's common to add some "politeness" words, like, "I'm sorry, with whom?" Or to rephrase to explain the problem, like, "I'm sorry, I didn't catch the name."
Among friends, in casual speech, the vast majority of native speakers, without so much as a blink, would simply respond
"Who with?"
On the other hand, in very formal (and academic) writing
With whom did you want to speak?
Or slightly less formal
Whom did you want to speak with?
Yes. You can end a sentence with a preposition. Yes, you can use the subject pronoun "who" even when prescriptivists tell you to use the object pronoun. You're speaking to a friend, not to a poet laureate. Whom may not be obsolete but it's certainly closer to death today than it was sixty years ago.
See Ngram
who do you live with (blue line) with whom do you live (red line)
