I was wondering whether there is a more common or 'best' way to define/think about the set of all complex number in complex analysis? The first way I can think of is:

$\Bbb{C} \triangleq \Bbb{R} \times \Bbb{I} = \{(a,b) \mid a\in\Bbb{R}, b\in\Bbb{I}\}$

Or is it more useful, down the road, to think of $\Bbb{C}$ as the set of all vector sums:

$\Bbb{C} \triangleq \{\vec{a}+\vec{b} \mid \vec{a}\in\Bbb{R}^1, \vec{b}\in\Bbb{I}^1\}$

and just remembering that you cannot add members/1-D vectors of $\Bbb{R}$ to members/1-D vectors of $\Bbb{I}$.

I guess another question would be how do you guys think about them?