Allow to differentiate autocomplete widgets if there are more than one per page.

I have a page with two search fields.
The fields have different length and thereby the autocomplete widgets must be different length too.

I have seen that an autocomplete wirdget for each field is added at the bottom of the page, but they are all identical and there ist no way to distinguish them.

It's possible to differentiate the widgets, so that they can be styled individually?

Thanks in advance.