After reading the following post (What is the "standard basis" for fields of complex numbers?), I tried to confirm that the suggested standard basis for $\mathbb C^2$ over $\mathbb C$ spans all of $\mathbb C^2$. The standard basis suggested by several answers is the set: $\{ (1,0), (0,1) \}$.
I played around with this for a while...and just wanted to make sure that my understanding is correct.
Before moving on, consider the following color coding scheme...as things will get messy. Let $\color{blue}{blue}$ represent an object that is an element of $\mathbb C$ and let $\color{red}{red}$ represent an object that is an element belonging to $\mathbb R$. Continuing...
When people say the set $\{ (1,0),(0,1)\}$ forms the basis for $\mathbb C^2$ over $\mathbb C$, they are, in fact, saying that $(1,0)$ and $(0,1)$ are both $ \in \mathbb C^2$. Using our color notation, we would say $(\color{blue}{1},\color{blue}{0}) \in \mathbb C^2$ and $(\color{blue}{0},\color{blue}{1}) \in \mathbb C^2$. Given that each one of those blue objects belongs to $\mathbb C$, and recalling that a complex number is an ordered pair of real numbers, we could then rewrite the basis accordingly:
$(\color{blue}{1},\color{blue}{0}) = \big((\color{red}{1},\color{red}{0}),(\color{red}{0},\color{red}{0})\big)$ and $(\color{blue}{0},\color{blue}{1}) = \big((\color{red}{0},\color{red}{0}),(\color{red}{1},\color{red}{0})\big)$.
For further clarity to the readers:
$\color{blue}{1} \mapsto (\color{red}{1},\color{red}{0})$ and $\color{blue}{0} \mapsto (\color{red}{0},\color{red}{0})$
Is this the correct understanding of what is meant when people say $\{ (1,0), (0,1) \}$ is a basis for $\mathbb C^2$ over $\mathbb C$?
Thanks!