Two national courts actually, there's the Federal Court too.
Then you have all the specialised courts, like the Family Courts, the Environmental Courts, the Court of Disputed Returns and many others that I can't remember off the top of my head.
And before the 1980's, you could technically appeal a high court decision to the Privy Council in the UK in certain circumstances, although I don't think that was ever used.
Talking about anachronistic ties to the UK, Australia should do something about all the UK citizens on the electoral roll. There are hundreds of thousands of non-Australian citizens (i.e UK citizens who arrived before 1984) who can vote in Australian elections. Really if you can’t decide if you want to be an Australian citizen after being here for 30 years then you really should stop voting in Australian elections!
Unless something has changed recently, a citizen of a Commonwealth nation (e.g. Australia) can vote in UK elections without being a citizen, as long as they are a resident. Maybe it's about reciprocity?
Often a court is the same judges as another court, so the proliferation can be illusory. For example, justices of the High Court recently sat as the Court of Disputed Returns in that WA Senate cockup.
Then you have all the specialised courts, like the Family Courts, the Environmental Courts, the Court of Disputed Returns and many others that I can't remember off the top of my head.
And before the 1980's, you could technically appeal a high court decision to the Privy Council in the UK in certain circumstances, although I don't think that was ever used.