autocomplete - functions for full-control?

https://editor.datatables.net/reference/field/autocomplete

I see I can only provide a string or bool to ajax and I can provide a function to ajaxData to "provide extra data parameters".

But for example how could I instead call my own custom endpoint and use query string only? For example: /api/entity/autocomplete-endpoint?search=hello%20world

ajax really needs to accept a function that can give me full control over the request and response. Unless it does and im missing something?