To elaborate on what TuxyDoh stated; items and equipment physically (as far as the game goes) on your characters or in your inventory will carry over with you through Cross-Save.
So if you collect the Deluxe items on PS4 and then go to play on PC, the items will be with you and should be usable as long as you have them on you.
However, if you then proceed to dismantle/delete them, you will not be able to pull them from Collections while still playing on PC; you'd have to log back into the Playstation to grab them from Collections.
So you do not have to purchase the Deluxe on all platforms unless you specifically want those extras available to grab on all platforms. They'll still be usable on all platforms you play on, but you'll have to specifically collect them on the platform you actually own the license on first (and each time you want to grab them again).
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That is definitely.. Well not perfect but still very valid and easy to follow. What about the seasons on the other hand?
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Edited by ADragonWithin: 10/9/2020 2:45:12 AMSeason Passes are applied to the save data directly, so purchasing the Deluxe version will give your characters the Season Pass unlocks which will carry over with Cross-Save. You just need to make sure to log in on the platform that you purchase the Deluxe on in order to get them applied to your characters, then you're all good and it will travel with them wherever you play them.