Neutralizing MAY not be necessary if its acid is completely depleted... but since I have no way of knowing for sure, I always neutralize.... well, not always, I have a couple of stocks I didn't neutralize, and they turned DARK within just a few years. One is nearly black.
I have found that baking soda keeps it about the same color as before neutralizing. I've only neutralized with ammonia once, and it also keeps the color the same. Usually, I will neutralize with lye, since they didn't have baking soda, and probably didn't use ammonia 250 years ago.
Lye will make the color more reddish, and I find that finishing this with linseed oil makes the same dark reddish oiled finish that I see on certain original guns. Many other old guns are pretty much bright orange, though, and they usually appear to have been finished with shellac/seedlac to fill the grain and some kind of varnish on top, and not finished with linseed oil.