SQL Server: how to write an alter index statement to add a column to the unique index?

You cannot alter an index – all you can do is

1. drop the old index (DROP INDEX (indexname) ON (tablename))

2. re-create the new index with the additional column in it:

      CREATE UNIQUE NONCLUSTERED INDEX (indexname)
      ON dbo.YourTableName(columns to include)
   

The ALTER INDEX statement in SQL Server (see docs) is available to alter certain properties (storage properties etc.) of an existing index, but it doesn’t allow changes to the columns that make up the index.