To hide mobile number when calling other people in Australia is as simple as it is in the UK but instead of adding 141 at the beginning you add 1831.
So if the number you want to call is 0481123nnn you would call 18310481123nnn
You could also request that your mobile phone provider blocks your phone number permanently but if you just want to try an get someone who is ignoring your calls to pick up or just don’t want someone your calling to have your number, you can simply prefix the number you want to call with 1831.
Note: I do not believe this works for text messages only when actually making a voice call.
Show your mobile number if you have it blocked by your provider
If you have got you provided to blocked your number permanently from showing you may be able to overwrite this by adding 1832 to the front of the number to instruct your number to be shown to the receiver.