Indexes are generally a mapping from the key to a row location (or sometimes to the primary key of the row). You can compress the index itself, but it's usually accessed randomly, so it might be detrimental.

Depending on the database, you might have compressed pages (sets of consecutive rows), rows, or columns in a row. Postgres does the latter (it's called TOAST) where the large columns are compressed and/or stored in a hidden table. So you first search the index for the key, figure out where your row is, read the row from the table, then, if needed, you read the columns you need from the TOAST table and uncompress them.