The search function in a view works on the first sorted column of the view.
If you have "dynamic" column sorting (user can click to re-sort view on another column), then the search <should> work on the new column.
If you expect to be able to "move" the search column without re-sorting the view, I'm afraid that is not possible. There is no parameter you can set anywhere to designate which column is supposed to be used for the search.
Pascal.