No, you can ship your electronics items. There is no duty payable on them if they are older than 12 months (or it might be 6 months old - I can't remember the exact figure). You will need to purchase an adaptor though as the Australian 3-pin plug is slightly different to the EU version.
You will also need to make sure that any items shipped over are thoroughly cleaned and free from dirt.
Your relocation agent will also be able to provide you with the list of prohibited items that typically will not be allowed in by quarantine.