Global search currently returns results from the entire moodle website.
This may return a great number of results, many of them may be irrelevant or distracting in a specific course or category setting.
A good feature would be if it could be limited to a specific category or course.
This limitation should be either defined by course/site admin/teacher when setting up a global search block instance (in a specific course or the front page, or by the user at the time of search if permitted/activated by admin, perhapds through the user interface of the advanced search.
For even better results, I also suggest adding other search parameters to the interface, such as time when document was created, type of document (doc, pdf, ppt, etc...).