"Widzieć" takes *accusative (widzieć kogo/co), therefore 'mnie' is the only option here.
Thanks mihxal for correcting me, I thought I finally remember the English names for corresponding Polish cases by heart, but I was wrong.
Only in Dative (the only case where 'mi' exists). "Mnie" used in Dative gets the stress, shows a contrast between 'me' and someone else. "Mi" is the standard, non-accented form and cannot be put neither at the beginning or the end of the sentence. Therefore you can say "Bardziej podoba mi się Ania" or "Mnie bardziej podoba się Ania" (I find Ania more attractive) and both will be correct. The second sentence shows contrast: my imaginary interlocutor likes another girl more, and I say that I think that Ania is prettier.
"Mnie" works also with Genitive ("Nie ma mnie tu" - "I'm not here", literally "There's no me here"), Accusative ("Ona mnie kocha" - "She loves me"), and Locative ("Oni rozmawiają o mnie" - "They are talking about me").
I think you may understand enough Polish to try and check the differences here.