Hi Andrew

you can either use the standard search of odoo website and extend

  @http.route('/website/snippet/autocomplete', type='json', auth='public', website=True)
def autocomplete(self, search_type=None, term=None, order=None, limit=5, max_nb_chars=999, options=None):

        """

        Returns list of results according to the term and options
....

    and the view website.search_text_with_highlight

or craete a module that has a snippet to be used in the website (header or somewhere else,,)

where you can controll more, but you need a JS, controller, view etc

This is an example of search in Blogs, categories, product, etc, and for each model, example product the search includes: name, barcode, INCI, and other custom fields

Hope this helps