Which rings $R$ containing (as a subring) the complex field $\mathbb{C}$ are, as a vector space over that field, isomorphic to $\mathbb{C}^2$?
In other words: what are the two-dimensional unital associative algebras over $\mathbb{C}$? Note that we do not a priori assume $R$ to be a commutative ring.
In general, let $k$ be any field. If $R$ is a $2$-dimensional $k$-algebra, it must have a basis $\{ 1, x \}$ where $x$ is not a scalar multiple of $1$; in particular, it must be commutative, since it is generated by $x$. Since it is $2$-dimensional, $x^2 = ax + b$ for some $a, b \in k$, and it follows that $R \cong k[x]/(x^2 - ax - b)$. The isomorphism type of $R$ is now controlled by the possible types of monic quadratic polynomials over $k$. There are three cases:
$x^2 - ax - b$ has two distinct roots in $k$. In this case $R \cong k \times k$ by the Chinese remainder theorem.
$x^2 - ax - b$ is irreducible over $k$. In this case $R$ is a quadratic field extension of $k$.
$x^2 - ax - b$ has two repeated roots in $k$. In this case $R \cong k[x]/x^2$.
If $k$ is algebraically closed then the second case never occurs and we conclude that there are exactly two isomorphism types of $2$-dimensional $k$-algebras.