Note: this question is inspired by someone (a teacher long time ago) who drew and labeled the two axes of the standard complex plane as $\mathbb{R}$ and $i\mathbb{R}$.
Let $i\mathbb{R} = \{iy| y\in \mathbb{R}, i \text{ is the imaginary unit}\}$
Does it make sense to write the complex plane $\mathbb{C}$ as:
$\mathbb{C} = \mathbb{R} \times i\mathbb{R}$?
Doing so, gives me that,
$\mathbb{C} = \{(x,iy)| x \in \mathbb{R}, iy \in i\mathbb{R}\}$
But then again, the complex plane is usually defined as:
$\mathbb{C} = \{z| z = x+iy, x \in \mathbb{R}, y \in \mathbb{R}\}$
How to resolve these two seemingly different definitions?
It makes perfect sense to define $\mathbb{C}$ in terms of $\mathbb{R} \times \mathbb{R}$, with addition defined by $$(a, b) + (c, d) = (a + c, b + d)$$ and multiplication defined by $$(a, b)(c, d) = (ac - bd, ad + bc).$$ Structurally, this is absolutely identical to any of the other definitions of complex numbers. If you're happy to abuse notations and notate such pairs of the form $(a, 0)$ by the real number $a$, and denote by $i$ the element $(0, 1)$, then $(a, b) = a + ib$ for any $a, b \in \mathbb{R}$.
So, how do we resolve the difference between this and other definitions? Well, we take the view that it doesn't matter. Whether you view complex numbers as matrices of the form $\begin{pmatrix} x & -y \\ y & x \end{pmatrix}$, or as elements of the splitting field $\mathbb{R}[x]/\langle x^2 + 1 \rangle$, or as ordered pairs from the definition above, the fact is that all of them produce an identical structure, and it is very simple to change between them in a way that respects addition and multiplication.
(FYI, I didn't define $\mathbb{R} \times i\mathbb{R}$, because, before defining complex numbers, it's not clear what $i$ actually is. Sure, you can say it's the square root of $-1$, but that only shows the property it's supposed to satisfy, not what it really is, or what context it can be well-defined.)