When PgAdmin III displays a list of databases, a database in the middle of restoring looks just like any other one. How can I determine if the restore has completed or not?
If by restore you mean pg_restore command in progress you cannot see that directly from pgAdmin. What pg_restore does in fact is execute simple CREATE TABLE, INSERT or COPY commands that differ in no way from normal commands. What you can do is you can open the Server status window. If you know where the command is executed (IP address) or if there is nothing else connecting to the database you can check if there are open connections to the database. If there are no open connections the restore has finished. If you can't deduce the info from connections you could look if there are any transactions (no transactions for some time = restore finished).
It would be simpler to get this information if you had access to the place where the command is executed.