The simple search is configured to work in conjunction with a content view. This section describes the document type and layouts used in the default simple search.
Simple Search Content View
The simple search content view is named simple_search and can be overridden see the contribution at search-contentviews-contrib.xml.
Quick Search Box
Since 5.8, the simple search box is shown thanks to an action. To customize this box, the corresponding action contribution can be overridden. See https://explorer.nuxeo.com/nuxeo/site/distribution/latest/viewContribution/org.nuxeo.ecm.platform.actions--actions
Suggestions
It takes over the simple search by disabling the action showing the simple search box, and adding the suggester one. See https://github.com/nuxeo/nuxeo-jsf-ui/blob/master/code/nuxeo-platform-suggestbox-jsf/src/main/resources/OSGI-INF/suggestbox-actions-contrib.xml.
Suggesters can be contributed to this search box. See the contribution at https://explorer.nuxeo.com/nuxeo/site/distribution/latest/viewContribution/org.nuxeo.ecm.platform.suggestbox.core.defaultSuggesters--suggesters.