Extending default search for name to consult a custom field
I can't think of a way to easily alter the search in all cases (quick search, autocomplete, normal search...

What we usually do when we have to store organisational units is to create a new contact type (department) and store together the org+"|"+dep as the organisation name. 

if this is a matter of making it easier to input the departments in that format or tweak some reports, you can still keep a separate field for the department and alter the organisation name via the hook, or split the organisation name on the "|" to separate the university from the department.

From a data modelling point of view, it won't win any beauty contest, but does the job. and it's more or less what's happening with the individuals, where first and last names are separate columns that are concatenated into the sort_name and display_name.