Conditional Formatting with multiple conditions

I've attached a de-identified version of the workbook. As you can see, column C should either have a date, or simply be blank. The far left hand column has the name of a clinician and underneath are clients. So, if there is a break between clinician names, there might not be anything in that particular row. I'm sorry that this is probably a confusing spreadsheet if you aren't used to working with it