First, ideally searching would open a search window that could be recalled once clicking away by using the browser's back button.
Second, the search results would be a customizable table or card view.
As a sales rep, I'd like my search to open in a window, so that I can return to my search results if I click on the wrong result, without re-searching.
As an admin, I'd like to customize the fields shown for search results. I'd like to do this from a search layout, not a field option, so I can see exactly the way the search will appear for users and design an optimized view.
This would be an incredible timesaver and empower the team to search using custom fields. This is an absolute must - product team, you know what to do!
I also agree with the noted benefits of such a feature. We have custom objects that are crucial to our searches. The default options are not always ideal based on additional but necessary information listed for a contact.