This is a rather inconcrete question and i am hoping for different answers:
The topology on $\mathbb{Q}$ and $\mathbb{R}$ is natural in being the order topology of these ordered fields.
Endowing $\mathbb{C}$ with the topology of $\mathbb{R}^2$ is the key to all those nice results of classical complex analysis and the important fact, that a function $\mathbb{C} \to \mathbb{C}$ is differentiable if and only if it is a conformal (locally angle-preserving) map. So the theory of holomorphic functions is more on geometry of $\mathbb{R}^2$ than on the properties of $\mathbb{C}$ as a field, I think.
Now suppose, the existence of an algebraic closure of $\mathbb{Q}$ would have been discovered before the invention of $\mathbb{C}$. It is hard to imagine that someone was like "lets endow $\mathbb{Q}[\sqrt{-1}]$ with the product metric of $\mathbb{Q}^2$ and embed its algebraic closure into its metric completion!"
Because there are so many other choices: Choose $\sqrt{2}$ instead of $\sqrt{-1}$; endow $\mathbb{Q}[\sqrt{-1},\sqrt{2}]$ with the product topology of $\mathbb{Q}^3$; etc.
My question is: Why does $\mathbb{C}$ deserve the euclidean topology of $\mathbb{R}^2$ and are there other choices?