Elideb: Gunslinger (at least in Win10) will crash right away by default unless run from a user account with admin permissions, or executed as administrator. You won't even get to see the publisher/dev videos at the beginning.

To avoid this, before starting the game, edit the file goglog.ini found in the game's installation folder. Replace "admin" in the following lines with your windows user name:
Dir_0=C:\Users\admin\AppData\Local\Techland\COJGunslinger

Dir_0=C:\Users\admin\AppData\Local\Techland
Dir_1=C:\Users\admin\AppData\Local\Techland\COJGunslinger\out
Dir_2=C:\Users\admin\AppData\Local\Techland\COJGunslinger\out\save
Dir_3=C:\Users\admin\AppData\Local\Techland\COJGunslinger\out\profiles

After that, you can start the game with no problem.

If, however, you've already executed the game, those values are already stored in the Windows Registry, and the wrong files might have been created.

First of all, go to the directories indicated above for your personal account (C:\Users\[username]). There, you should find a techland directory. Remove it, because it will not contain the right files.

If you've executed the game as admin and managed to start it, the registry will contain the wrong information.
After that, open the windows registry (regedit) with admin permissions (right button, Run as admin). Navigate to
Computer\HKEY_LOCAL_MACHINE\SOFTWARE\WOW6432Node\GOG.com\Games
and locate the CoJ Gunslinger folder. In it, there will be a variable named savegamefolder, which will contain the admin directory. Edit it and replace admin with your windows account. It should be something like C:\Users\[UserName]\AppData\Local\Techland\COJGunslinger

After that, you should be able to start the game, which will then create the files in the right place, and no longer require admin permissions to run.

Bonus info: if you've run the game as admin before, your saves will be in C:\Users\admin\AppData\Local\Techland directory. You can copy and paste them into your user directory (but will need admin permissions to do so).

@GOG: There's probably some variable that can be used in goglog.ini to indicate "the user's account name", so this is avoided from the get go. goglog.ini should be updated to use that.