The proof of Cauchy's theorem in these notes http://people.math.gatech.edu/~cain/winter99/ch5.pdf rely on the concept of homotopy. But it seems to me that the proof did not use any property special to the complex plane.
My guess is that there doesn't generally exist a homotopy between curves in the real plane. But it is easy to come up with simple homotopies in $\mathbb{R}^2$ (using vectors).
In what way did this proof use a property exclusive to the complex plane?
The Cauchy integral theorem says that if $f$ is an holomorphic function on the bounded simply connected domain $D$, whose boundary $\partial D$ is piecewise differentiable, and $\gamma:[a,b]\to D$ is a closed path parameterizing the closed curve $\partial D$, then $\oint_\gamma f(z)dz=0$.
The proof involving a homoptopy just exploits the definition of a simply connected domain, namely $D$ is simply connected if any closed path in $D$ can be continuously shrunk to a point in $D$. Moreover the computations of the derivative $I'(s)$ in the file at your link, are based on the hypothesis that $f$ is holomorphic in $D$, hence there exists the derivative $f'(\gamma_s(t))$ along each path $\gamma_s$.
The notion of homotopy between two continuous paths is not specific to paths in $\mathbb{C}$, but in any topological space, in particular in $\mathbb{R}^2$, as well.