Do I need a product key for AudioNote for Windows?
AudioNote for Windows is free to use as a viewer for .audionote files created elsewhere. You will only need a product key if you wish to record on Windows (to purchase, go to Help->Upgrade).
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Do I need to purchase AudioNote separately on iOS, Mac, Windows and/or Android?
The various app stores process app purchases separately, so if you wanted to record on each device then yes, you would need to purchase for each platform separately. AudioNote is free to use, however, to view notes recorded elsewhere.