Incredibly helpful post

Alpha Centauri and Alien Crossfire work 100% ok on AMD 14.4 catalyst drivers (Windows 8.1) AMD HD7950. All graphics work without any glitches at 2560x1440 resolution.

After searching some, I found this link:

www.weplayciv.com/forums/showthread.php?7019-Alpha-Centauri-Alien-Crossfire-Graphics-not-working-properly

I take no credit for this, I merely found it, but it fixed the rainbow graphics for me on my laptop. My desktop machine has an HD7700 but didn't have any problems with the most recent drivers.

The solution posted there is:
The menu still has some horizontal lines, but the in-game graphics and pop-up menu's are just fine. So, thanks alot! I'll just summarize my solution down here for people who have the same problem (Windows 7):

First, type '%appdata%' in the search bar of, for example, 'My Documents'. The next step is to go to the Local directory of AppData and select Virtual Draw, then Program Files and then Firaxis Games. This is the directory where you should be:

AppData\Local\VirtualStore\Program Files (x86)\Firaxis Games\Sid Meier's Alpha Centauri

Once your there, open your Alpha Centauri.ini file (make sure you have started the game once before you open this file) and type 'DirectDraw=0' somewhere under [Alpha Centauri]. It will look something like this:

[Alpha Centauri]
DirectDraw=0
ds3d=1
eax=1
FastUnitAnim=0
... (etc.)

That should solve your problem! For me it wasn't unnecessary to add 'video mode=800' (depends on your native resolution) but you can also try that if the first doesn't work. I also set 'ForceOldVoxelAlgorithm=1' but I do not believe that has any effect on graphics (confirmation?)

Once again, thanks Petek!

EDIT:
I fixed the horizontal lines aswell by forcing my laptops resolution to 1024x768. If the resolution of your laptop can't go higher then 1024x600, there are guides on the internet how to force this resolution on your machine (but watch out, there are high risks involved if this is not done properly!).