Well....yes and no.
If you have Civ3 vanilla, which I am assuming you do, it will not 'upgrade' to Complete. Complete is not a version of Civilization III, it is the 'Complete' collection of all three iterations of Civ III, containing the fully patched versions of Conquests, Play the World and vanilla, all of which install at the same time. However, this install only installs a shortcut to Conquests. No big deal, you can always add your own shortcut to PTW or vanilla.
It will not look for your current copy of the game and try to fix it. Each game installs into it's own folder with their own graphics, save folder and game engine.
You can install Complete and still play vanilla.
Conquests can open games saved in Vanilla or PTW, but the reverse is not true.
If you do decide to get Complete, I would suggest uninstalling your current version of Civ III (keeping the saved games intact, of course!), and then install Complete. Different versions were made by different distributors and use different CD-checking schema and they don't always match up.
If you have the latest patch of CivIII Conquests, you don't really gain anything by getting Complete. With vanilla or PTW I think it is worth the cost.
My first version was Civ III Gold, a combo package with both vanilla and PTW. But I really enjoy the Succession Games and most of those are in Complete. Once the SGOTM went to Conquests only I had to upgrade. Now Conquests is all I play, when I play.