I wouldn’t know if that’s necessary or not, since it was a playthrough with everything I already had. But one thing is certain: your magic level has to be high, around rank S, and your alignment must be Chaos. She has to challenge you and you have to defeat her; she dies, and if you go to her sister afterward, I think she gives you resistance to darkness. Your Chaos level needs to be something like 90% or higher to obtain Archmage.
I just wing it in my chaos slut mage run and just get the title near in the end of the run.
Kind of still hard to gauge yeah I think the requirements are now a bit solid to get the
Archmage title/skill status.
1. High Chaos(?)
2. Learn every possible magic on Carmilla and maybe other ones that aren't in her inventory
3. INT going around S rank(70)
4. Carmilla initiate a dialogue of duel to death via learning
Ancient Script Analysis
5. Acquire the Staff of Io(?)
I'll test it in a fresh new game+ if its possible to get it even with few of the requirements
Edit: I get the Archmage in a fresh NG+ with fair amount of Chaos around 80%, I think the key points are:
1. Learn all of the Spells she have.
2. Learn
Ancient Script Analysis and
have her initiate the duel.