Yup, you’ll notice the only thing distinguishing C from R^(2) is that multiplication. That one definition has extremely broad implications.

For fun, another definition is in terms of 2x2 matrices with real entries. The identity matrix

1 0
0 1

is identified with the real number 1, and the matrix

0 1
-1 0

is identified with i. Given this setup, the normal definitions of matrix addition and multiplication define the complex numbers.