Intuition behind the usual topolgy on Complex numbers

Question

I feel that the concept of complex field was a result of natural intuition seeking for an algebraic extension (field extension) for the field of real numbers, the smallest field containing both $\mathbb R$ and the potential solution for $x^2+1=0$. But the definitions of conjugate, Modulus (norm), Argument, etc., of complex numbers are exactly matching with Euclidean space $\mathbb R^2$, My question is about the necessity behind equivalency of topology induced on $\Bbb C$ with the usual topology on $\Bbb R^2$, More explicitly, what was the intuition behind the topology on $\Bbb C$ to chose the usual topology on $\Bbb R^2$? Is it only due to the dimension of $\Bbb C$ over $\Bbb R$ being a vector space?