Usefulness of alternative constructions of the complex numbers Complex numbers $\mathbb{C}$ are usually constructed as $\mathbb{R}^2$ together with a suitable multiplication.
But this is not the only possible way, one can get to the complex numbers. One alternative would be via $\mathbb{R}[x]/{\left< x^2+1\right>}$, but I have never seen this one be used in practice.
When visualizing complex number one usually uses the fact tha
  and then visualizes what happens in $\mathbb{R}^2$.
,    
My  question is: 1) Are there any instances of theorems or definitions involving complex numbers, were one is better off* using a different construction than $\mathbb{R}^2$ ? (You may also answer with the contrapositive and describe a situation were using $\mathbb{R}^2$ makes everything worse.)
2) A special case of 1): Are there known constructions of complex numbers other than $\mathbb{R}^2$  that can also be given a geometric/visual interpretation ?
One reason the construction via $\mathbb{R}^2$ is more popular than $\mathbb{R}[x]/{\left< x^2+1\right>}$ is surely due to the fact that by this interpretation the complex numbers (an also their arithmetic operations) have a geometric meaning - whereas visualizing equivalence classes of polynomials is not so easy.       
*this could mean anything that helps our understanding: For example it could mean that the proof becomes conceptually easier, the situation is easier to visualize etc. 
 A: Original poster misunderstands relationship between $⟨1,i⟩$ with $i^2=-1$ (said $\mathbb{R}^2$) and $\mathbb{R}[i]/{⟨i^2+1⟩}$. These definitions of a ring extension are not fundamentally different, but the latter starts from a property that the extension must acquire (namely, a new $\sqrt{-1}$ element), whereas the former shows its vector space structure over the ground field ℝ explicitly, in the form of standard basis. Original poster “never saw this $\mathbb{R}[i]/{⟨i^2+1⟩}$ be used in practice” only because degree-2 polynomials on $i$ are immediately reduced to lesser-degree polynomials, yielding aforementioned 2-dimensional vector space.
Yes, there exists another broadly known representation of complex numbers, namely by two numerical parameters: non-negative modulus (absolute value), and angular argument. It is related to “$\mathbb{R}^2$” is the same way as polar coordinates are related to Cartesian ones. Where and why is it convenient? Complex numbers are a handy metaphor for linear transformations of Euclidean 2-vectors that preserve (signed) angles. Such transformations include rotations as complex numbers of the modulus 1, dilations (homotheties) as real numbers, and all their compositions as products. Saying our transformation is $r\exp(iθ), r≥0$ answers two questions:


*

*What happens to vector lengths? (are multiplied by r)

*By which angle are all vectors rotated? (θ)


Using “$a + bi$” parametrization indeed gives Cartesian coordinates of the image of one of standard basis vectors, some expressions of orthogonal projections for a general Euclidean vector and its image, but is far less sensible geometrically.
