What are "sys.extended_index..." indexes?
Extended Indexes are full text indexes, XML indexes, and Spatial indexes.  Reason, the 3 types of indexes are better handled outside the B+ tree index.

I know you do not care to correct this but, did you recently upgrade?  Try changing the compatibility level if it's not SQL Server 2014 and run CHECKDB again

You can identify the table object for the extended index error by using the below code. You may need to tweak the name in the where clause if I have it wrong.

SELECT 
    OBJECT_NAME(PARENT_OBJECT_ID) AS table_object,
    name,
    create_date
FROM sys.internal_tables it
WHERE INTERNAL_TYPE = 207
AND it.name = 'extended_index_1066482673_384000'


Hope this helps!