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I added languages spoken to the custom fields.

I’m not really sure where you added it or what you mean. Do you mean the custom fields on the person single edit screen? If yes, you have to edit the single-person.php file and add it in the code, more about this WordPress core function you can find here:

https://codex.wordpress.org/Custom_Fields

But I think it’s easier for you to add it in the content…

Also, is there a way to make search function use other person fields as well?

No, without massive changes in the PHP files it’s not possible. Of course, it’s a high complex function…