This is the first time my Kaffir lime has had new growth and it's almost black. I guess its a deep dark purple or red. Is this normal? I'm assuming so, kind of like how lemon's new growth is light purple.

Yes. Have a look at this picture.

My Diamante citron has the same growth.

Yes its normal mine does this too.
After a week or so it goes light lime green.

Usually cold temperatures induces violet leaves, when in may cold comes, the growth of my lemon trees get deep violet nearly black, even the new flower buds look black. My Key Lime also get's puple new gowth and my kaffir got this coloration, too.
I first tought this is of sunlight, but it seems that cold and sunlight induces a more intense coloration than cold alone. Seems to be a protection of the juvenille shoots against the harsh sun rays - guess about antocyanine pigments in the growth