Medicare recipients have an option of buying private insurance outside of Medicare, often called “Advantage” plans. These plans are offered by private companies that contract with Medicare, and often include benefits not provided by regular Medicare, such as vision, dental, etc.
This type of private insurance plan must conform to the rules of Medicare--such as limiting charges for certain services--but is administered by Blue Cross, United Health, or other private companies. Patients still have the rights and protections provided under original Medicare.
For more information, go here: https://www.medicare.gov/sign-up-change-plans/types-of-medicare-health-plans/medicare-advantage-plans