In the usual definition of the field $\mathbb{C}$, as $\{(a,b):a,b\in\mathbb{R}\}$, the field $\mathbb{R}$ is not exactly a subset of $\mathbb{C}$, but only an isomorphic copy of the subfield $\{(a,0):a\in\mathbb{R}\}$. I am wondering: can we define $\mathbb{C}$ as 
the set 
$$
\mathbb{C}=\mathbb{R}\cup\{(a,b):a,b\in\mathbb{R}\ \text{and}\ b\neq0\}\ ?
$$
In this case, how to define the operations?

$\mathbf{Remark}$: We know that the operations of $\{(a,0)\}$ can be extend to 
$\{(a,b)\}$. Since $\mathbb{R}≅\{(a,0)\}$, can we prove that the operations of $\mathbb{R}$ can be extend to $\mathbb{R}∪\{(a,b):b≠0\}$?