How can I turn my muffin batter into the natural blue color from blueberries?

The story is more complicated than SAJ tells it. Blueberries, like many other purple foods, are colored by a pigment called anthocyanin. It changes its color from red at very low pH to real blue at very high pH. At the blueberry's natural pH, the color is a purple with more red than blue in it.

What you can do is to juice some blueberries separately, then add lots of baking soda to the juice. It will turn a dark blue. You can then mix it into the batter.

The problem is that you will be changing the taste of the muffins, and some of their rising behavior. If you use little enough blue-turned juice to not make a major change, you will get dirty greenish muffins, as the batter itself is yellow-beige, and that will mix with the blue. If you use enough to color them, you will have pretty display muffins (provided they don't stay flat because you turned the batter too alkaline for the baking powder to work) which will taste like soap.

Personally, I will stick to food coloring. "Natural" coloring is an unachievable utopic for most foods, as natural pigments are finicky and almost never concentrated enough to color a food flavor with them. The only time they work is when a food is made predominantly from the coloring food, e.g. blueberry sorbet is indeed red-purple from the blueberries contained in it.

This is an example of naturally colored shortbread cookie dough. The lavender dough was colored using elderberry fruit juice with lots of baking soda. It was very noticeable in the taste, I wouldn't do it again. Ironically, the slightly soapy taste fit a bit with the lavender aroma, it was just weird eating the stuff.

Blueberries, despite the name, are purple, not blue. The appearance of being bluish in color is due to iridescence, not pigment.

While you certainly could color the batter purple by pureeing some of the blueberries into it, it won't be blue. Think about the color or stain near the berries blueberry muffins you have eaten—it is a rich purple, not blue.

True blue is an exceedingly rare color in natural foods. The only even close to common food that is blue is blue cornmeal. You can certainly make blue corn muffins if you can obtain blue cornmeal.

Sadly, if you want blue muffins that are not based on blue cornmeal, you will almost certainly need food coloring.

I have found putting my blueberries, I use frozen, into the liquid for a few minutes before adding the liquids to the dry ingredients will give your muffins a nice blue color. The liquid will have a slightly purple look to it. The baking powder in the dry mix will then react with the blueberry juices and turn the batter a nice shade of blue.

Puree a few of the berries, say 1/4 cup, before adding to the batter. That shouldn't be enough liquid to mess up the recipe badly. Work quickly once you've got the puree in though, blueberries are acid enough (pH about 3.1) to start your leavening agent working immediately. That'll mean flat muffins if you don't pop them in a hot oven fast.