I had once defined $\mathbb{C}$ as $\mathbb{R}\times\mathbb{R}$ as a set, then I found it extremely uncomfortable when doing complex analysis or measure theory. So I changed its definition so that $\mathbb{R}\subset\mathbb{C}$.
Now, I just faced $\mathbb{H}$(Hamilton) and I'm hesitating to choose the definition for it.
Is it a good choice to define it as a set such that $\mathbb{C}\subset\mathbb{H}$? Or is it okay to define it as $\mathbb{R}^4$ (In this case, neither $\mathbb{R}$ nor $\mathbb{C}$ is a subset of $\mathbb{H}$)
And is $\mathbb{H}$ being used in analysis for computations?