So I understand the GOG version of the game comes with BASIC version of the patch that fixes a lot of the bugs and issues. My question is that if I start the game with the BASIC patch and complete it. Can I upgrade to the plus patch later and play the added content with my current character?

I'm confused: a new game means a new character so using the plus patch shouldn't be an issue since there won't be any old saves to cause any discontinuities.

Wait, do you mean can you go back and do only the added sections with the current character after completing the game? You MIGHT be able to reload an older save and do that, but it wouldn't surprise me if things don't work 100%.

At any rate I wouldn't recommend it, because the added content is in a number of places, which wouldn't be obvious while playing through without the plus patch. Of course you can look this up via guides etc, but personally I'd start a new character for a new play through. This game doesn't take a huge amount of time to play through, and it's worth trying out different character classes for your second pass through - although even choosing the same class may be considerably different due to the changes made (due to not only added content, but also differing abilities, slightly different way stats work, etc.).

Most probably you will break the game by this. To replay with the plus patch, just install this over your current game and then start a new character...