I noticed my neighbor down the street has a 2008-10 or so Chrysler Town and Country with the factory HID low beam headlamps. They look a bit of a bluish color.

Do these types of headlamps always look this way from new, or does the color shift, similarly to how MH lamps turn greenish over time? Since I do believe that the bulbs on these are a sort of metal halide type.

I've seen them go a pink-ish color but not green

I've seen them shift slightly over the span of their lives, I was just noticing recently that one of the HID headlamps in my father's car had a slight greenish hue, while the other had a slight blueish hue despite both being the same age and make. Sadly i can't get a pic since the blueish one just burnt out today.

Sure they can. Legal HID lamps are sodium-scandium metal halides. So, sodium loss will shift them to blue-green, and scandium loss will shift to orange-pink. I have seen both. Aftermarket lamps may contain whatever witch brew of halides and can show different effects)