Anthem Health Insurance headquarters is shown in Indianapolis, Feb. 5, 2015.

Health insurer Anthem is offering free credit monitoring after a major breach that may have affected as many as 80 million records, but customers should watch out for an especially insidious type of fraud: medical identity theft.

Anthem disclosed the hack late Wednesday, saying customer information that could have been compromised includes names, Social Security numbers, street addressesand the medical ID numbers found on customers' health insurance cards.