How do we Distinguish the Complex Numbers from Euclidean $\mathbb R^2 $? We know the two are of equal Cardinality $\mathbb c \times \mathbb c $, diffeomorphic as Real 2-manifolds, though I am not sure that Euclidean 2-Space is a Complex 1-manifold. The two are homeomorphic per the map $(x,y) \rightarrow x+iy $ I guess the Complexes are an algebra ( over the Reals) and the Euclidean Reals are not? I am also confused as to why/how z=x+iy is a single (Complex) variable: what "binds" x,y (both Real numbers) into the (single) variable $x+iy$?
Thanks.

You got it. $\mathbb C$ is an $\mathbb R$-algebra, while $\mathbb R^2$ is not (naturally speaking; c.f. Noah Schweber's comment). In fact you can define $\mathbb C$ as $\mathbb R^2$ with a fancy operation added onto it, namely $$(a,b)\star(c,d)=(ac-bd,ad+bc)$$
