You're talking about adding ingredients with completely new effects, yes? For example, maybe a few ingredients that would have Shock Damage that could be used to create poisons that inflicted shock damage?
The effects of Alchemy are all magic effects, listed in the Magic > Magic Effects category. They all start with "Alch". I imagine you could pretty easily add new effects that would use existing (but unused) types of damage or other effects like maybe Calm or something. Just look through some of the different "values" and such you can add or modify. You might even be able to create completely new effects through magic effects with scripts attached.
Yes...I was thinking about that...

I'm not sure if the game would actually recognize them via potion making, though. I'm guessing it would, because it seems like the way it works is that the game simply looks at magic effects on ingredients and if they match, they get added to the potion, but I can't be certain, there may be some degree of hardcoding to it. Shouldn't be too incredibly hard to test, though. Make a new magic effect, edit a couple of the existing ingredients (Items > Ingredients) that are easy to get in-game or that you already have in-game...
Oh...well...I would test it right through console commands.

In fact I was adding a couple of special ingredients and I wanted to see if adding new kinds of effects would be possible...

I was just puzzled because I saw no hint about how potions are mixed...
Go out and harvest ingredients like flowers, mushrooms, bug's, dragonflies, mudcrabs chitin. you can often see it's effects when looting. Find alchemy labs and make potions.
You can also loot recipes from necromancers.
That's what i know, probably someone knows more
Emmmh...I didn't meant that, but thanks notwithstanding...

Jashkar