I am trying to add new fields to the user search results in setup. For an example, I need to add Profile of the user. But I cannot find a way. Adapting the search results layout doesn't help in this case.
Any solution?
You are referring to Setup Search Results Page (Beta)
As you can see, the columns are fixed:
Name Type Object Last Modified Date Last Modified By These are fixed because the Setup search is searching across many different object types and hence ProfileId would not apply to a Custom Object, Field, Email Alert, Approval Process, ... The Setup Search is designed to find items in setup so you can further explore them.
To see the user's profile, you should go to Setup | Manage Users | Users and choose/modify a list view to show the ProfileId or use @DhananjayPatil answer for normal global search.
Goto Object Manager, Search for User Object. Navigate to User Object and there is an option for Search Layout, Edit the Search Layout and add the field "Profile" to Search Layout. Attaching screenshot for your reference.