No it does not turn soap brown. Even when I scorch mine with the lye the brown curds don't make a difference. Its not a really nice white like I get from batches made with water only. But it is a creamy looking white.
I've considered making my own almond milk. As far as I can tell the only drawback would be that it would actually cost me more to buy the almonds and make it myself. I plan to look more into costs later when I have a little more money saved up. I can't really tell you if the color changes depending on store bought or homemade almond milk. But the store bought is just fine to me.
This is a good pic of some soap I made with almond milk. Hope that helps.