This is an easy question to answer if you realize it depends:
If you are an independent professional and your patients will have you as a Facebook friend absolutely!
It is an implied endorsement of your practice. It enables them to connect with you and you them. Audiology is a personal service business; with a big emphasis on personal. So if they are your patients then why would you not want to friend them? This is the cheapest marketing available to you.
You may even want to have a personal and a professional Facebook account as your professional page might be different from a Facebook page you use with your high school friends. You could have one account as Richard Poage, Au.D. for your professional contacts, and another account as Rick Poage for your friends. This is not as weird as it might at first sound. You would present yourself differently at an office open house than at a class reunion at the lake. Your patients don't need, or probably want, to see you in your swimsuit with a beverage. Your classmates probably don't really care about your office promotion. You may even have someone on your staff update and monitor your professional page for you, just like they answer you phone and monitor the appearance of your reception area.
If you are not an independent professional and your patients belong to your employer, then no.
If you are working in a situation where the patients are not yours, but the hospitals, or the VA's, or some other group where you have no real connection with them as individuals but only as people your employer puts on tour schedule, then no you would not want to friend them. You have no real control over who you see or what kind of people they are. You have no control over why they come to you or what they may think of your employer.
In this situation you may want to even limit your personal page so most is only visible to particular people.
Rick Poage
907-229-3480