The domains of x and y both appear defined as the real numbers. But, the reals do not have a root for every non-constant polynomial defined in terms of x or y a

The domains of x and y both appear defined as the real numbers. But, the reals do not have a root for every non-constant polynomial defined in terms of x or y as the real numbers are an not algebraically closed field. The proposed concept attaches two extra (complex) dimensions to each two dimensional real point. This is close to visualizing R^4 by attaching an R^2 to each point of R^2, so in some sense the tangent bundle of R^2.