Why is |a+bi| defined as: $\displaystyle \sqrt{a^2+b^2$

Why is it not defined as $\displaystyle \sqrt{(a+bi)^2$

This would yield $\displaystyle \sqrt{a^2-b^2+2abi$

I understand that it comes from applying pythagorean's theorem to the complex plane but since proofs of pythagorean's theorem obviously involve only real numbers I guess it's just a convenient definition so that other results come out the way we want? Is that the idea of even defining the imaginary plane to begin with?