Well, there are two version of Complete around:
One contains all the necessary exe-files to start Vanilla or PTW in the respective folders. This version comes on three CDs and is for example included in the Civ Legacy box or the Sold Out Games package. If you have that version I'd say it should be possible to start PTW and play with another PTW-only installation. Don't unse the Conquest/Complete-exe to start your game, this surely won't work!
And then there is the other version - which for whatever crazy reason omits the Vanilla- and PTW-exe files!!! So you are technically limited to playing the Conquests-part only. However in the version that I have and know it's really just the exe-files missing, everything else is there - so if you have the chance to get those missing exe-files from anywhere, for example form your friends installation and the disk check fits your own game disk, you should be able to upgrade this incomplete and falsly advertised "Complete" version to a really Complete version.
It might be also necessary for your friend to update his PTW-version to the most recent version, but if that is done I see no reason why this should not work.