In summary it's not possible to add additional departments. The three default department (sales, lettings and commercial) are hardcoded deep into the plugins code as so many things depend on which department is set (i.e. the fields available and how they display on the frontend).
It is however possible to give the illusion of different departments on the frontend. Let's say, for example, we wanted to add a new 'department' of 'Students'. Properties would still be added as lettings properties with the 'Residential Lettings' department selected but we can use Marketing Flags to specify that they are student properties.
Step 1 - Add the Marketing Flag
Navigate to 'Property Hive > Settings > Custom Fields > Marketing Flags' and add a new option of 'Student'. Take a note of the ID of the marketing flag (available on the main page) as we'll need this in step 3.
Step 2 - Select the Properties in Question
When editing a property now you can navigate to the 'Marketing' tab and tick the 'Student' checkbox that's now available.
Step 3 - Filter these properties on the front end
You can now show just these properties by appending 'marketing_flag=X' to the URL (where X is the ID from step 1). So you're page URL to show student properties would be something like:
https://{site url}/(search page url}/?department=residential-lettings&marketing_flag=X