In private practice, you have the liberty to choose your doctor and to continue seeing the same doctor for as long as you wish. There are plenty of private clinics as well as private hospitals in Melbourne. They offer the whole range of inpatient and outpatient services. For example, I'm affiliated with three private practices including Northpark Private Hospital, Glencairn Consulting Suites, and Essendon Private Clinic. 

Many people find private services more accommodating than that of public services. You will always be seen by a senior doctor and you can change your doctor if you're not satisfied for any reason. But private services are not free. Medicare will reimburse most of what you pay, but you may still need to pay a bit out of your pocket.