[jacobolus]

Complex numbers are (isomorphic to) a certain subset of linear operators on a 2-dimensional Euclidean vector space which rotate and/or scale the vectors in the plane without skewing or anisotropically stretching them. (We usually also include a zero operator here; depending on use case we sometimes omit it (the “punctured plane”), or sometimes add a point at infinity (the “Riemann sphere”).)

If you want you can write them down as matrices acting on vectors in an orthonormal basis by matrix multiplication:

  [a -b]
  [b  a]

Or if you prefer you can consider i to be a unit bivector in the plane, with a complex a + bi acting on vectors by Clifford’s geometric product.

Or if you want you can write them using a length and an angle measure, and use high school trigonometry to figure out how they apply to vectors.

The difference between the real plane and the complex numbers is that for two vectors in the plane, the product is not a vector. (Indeed, if you use Clifford’s geometric product, then the product of two vectors is a complex number (scalar + bivector).)