People have proposed introducing a symbol for 2π before, most often τ. I like to go a step further and introduce a symbol for 2πi. I use pi with a dot above it, pronounced "pi dot". Pi dot can be defined as the period of the exponential function (which can be defined in terms of its Taylor series). Then 2π is pi dot / i, and π is pi dot / 2i. Of π, 2π, and 2πi, 2πi is probably the most natural, even though it's imaginary. I suppose that depends on the type of math you're doing though.

On a similar note, when doing quantum physics, I like to introduce h dot, which is i × h bar. There are tons of formulas where you either get i × h bar or -i / h bar, but these are just h dot and 1 / h dot, so this removes a little sign confusion and saves a little handwriting.

People will argue that real constants are more natural, but maybe they're not. Maybe radians are naturally imaginary, so if h bar is meant to have dimensions of energy time per radian, then it's better to use the imaginary h dot.