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I'm trying to transfer a save/character from one PC to another, and it seems the game doesn't support cloud saving, so I have no clue as to which file is the Charactor file, or save location, in order to copy over to my newer desktop PC.

Any help would be appreciated
A Google search for diablo 1 gog save file location returns, among other results, GOG Diablo 1 Forum: Save game location. The information there appears correct from what I remember.