Alien Crossfire (the expansion for Alpha Centauri) still appears to work perfectly here when run as administrator (I'm on Windows 8 consumer preview currently; haven't updated to release preview yet).
The Alpha Centauri.ini should be in the game's folder, not appdata, especially if you're running it as administrator - although I do not know if GoG changed where it is located - my version isn't from GoG. Things I added to mine to make it work:
At the beginning:
[Alpha Centauri]
DirectDraw=0
and at the end:
[PREFERENCES]
ForceOldVoxelAlgorithm=1
Obviously don't add the [Alpha Centauri] line if it's already there, just change or add the DirectDraw=0 line. The [PREFERENCES] section with ForceOldVoxelAlgorithm=1 was not there by default, however (unless GoG added it, since it's basically necessary on all modern computers AFAIK).
I also have all compatibility modes disabled, am not running it in lower resolution, etc, and it is appearing at my desktop resolution. However, running at the full desktop resolution with DirectDraw=0 does not work with SMAC (it was a new feature or bugfix in SMAX), unless GoG has backported it into SMAC or you have SMAX and are running terranx.exe instead of terran.exe. (What happens, and what you'll experience if the bug is still there in GoG's version of SMAC, is that part of the screen will be cut off and will show up as black, so you likely wouldn't be able to see the main menu at a normal desktop resolution with DirectDraw=0 using terran.exe instead of terranx.exe, unless GoG backported the fix from terranx.exe to terran.exe)
(What is DxWnd?)