I've been wondering if there is any use to defining a set that is isomorphic to $\mathbb{Q}^2$ (in the same way that $\mathbb{C}$ is isomorphic to $\mathbb{R}^2$).
I immediately see a problem with e.g. Cauchy's theorem (there's no $\pi$ available) and perhaps even defining a "analytical function" concept because there are series defined in $\mathbb{Q}$ that do not converge within $\mathbb{Q}$.
Still, I wonder if there's a way around these difficulties.
On
Yes, such a field exists. It's sometimes called $\mathbb{Q}(i)$. It's a degree two extension of $\mathbb{Q}$.
It has the property that some polynomials with coefficients in $\mathbb{Q}$ have roots in $\mathbb{Q}(i)$. For instance, $X^2+1$.
However, the complex numbers are algebraically closed—any polynomial with coefficients in $\mathbb{C}$ has roots in $\mathbb{C}$. $\mathbb{Q}(i)$ does not have this property: $X^2-2$ has no root in $\mathbb{Q}(i)$.
Also, the complex numbers are complete as a metric space, meaning that Cauchy sequences in $\mathbb{C}$ converge to elements of $\mathbb{C}$. $\mathbb{Q}(i)$ does not have this property, either, making it hard to analyze functions defined on this space.
People do indeed define the "complex rationals", i.e. the extension of the rationals by $i$, often cqlled the Gaussian rationals. The question is what you want to do with this.