In at least a few areas (i.e., those that I have happened across) when we have a need to capture a half-plane we do so by taking a semi-circle of radius $r$ in that half, and taking the limit as $r\rightarrow\infty$.

For example, the positive-imaginary half-plane for Jordan's Lemma, or the positive-real half-plane to construct the Nyquist Contour.

1. What motivates this model?

It would seem to me that in a 'Cartesian-like' grid, we should have to take a semi-square of side length $s$ (that is, an $s, 2s$ sided rectangle) in the limit $s\rightarrow\infty$.

2. How do we know the semi-circle construction is equivalent? (Or, why am I wrong, and the rectangular construction doesn't work?)

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I have a sort of sketchy idea from discussion in comments of @texasfloods' answer that I can't think how to formalise - that given the unit circle $x^2+y^2=r^2$ it seems intuitive that taking $\lim\_{r\rightarrow\infty}$ is identical to $\lim\_{\sqrt{x^2+y^2}\rightarrow\infty}$, the hypotenuse of the right triangle, and by a bit of hand-waving, "done".

Also from @pbs' comment below, it seems reasonable to say that for any open set $A$ of points enclosed by any 'contour shape' $\Gamma, \forall z\in A$ there exists a 'large-enough' construction of $\Gamma$ to enclose $z$.

I'm fairly convinced, but I'd still be interested in a more 'proper'/rigorous proof or explanation if anyone cares to offer one.