I try finding a way to parametrize in Complex Plane the path: $$ \left \{ z \in \mathbb{C} | \begin{cases} z = a+ib, (a,b) \in \mathbb{R}^2 , \\ |a| + |b| = 1 \end{cases} \right \} $$
Trivially, the result would be a piecewise defined path ie. a union of 4 lines.
While trying, I found that the parametrization in $ \mathbb{R} $ for the real valued function which would induce the same graph as this path doesn't hold in $ \mathbb{C} $.
In fact, the values $ a,b $ tend to be non-constant functions of one of another. And this situation shows that we need to choose another parameter.
For the new parameter, I would suggest using $ |z| $, thus over the couple $ ( \rho, \theta ) $ where $ z = \rho {e}^{2 \theta \pi i} $ in polar form.
But my calculations haven't led to any solution.
Thanks for your help
There are many ways of parameterising this path, it really depends on what properties you want.
Let $x(t) = \max(1-x,x-3)$ for $x\in [0,4]$ and $4$-periodic otherwise. Let $y(t) = x(t-1)$. Then $\gamma(t) = x(t)+iy(t)$ for $t \in [0,4]$ will suffice.
You could also try $\gamma(t) = {1 \over \|e^{it}\|_1} e^{it}$ for $t \in [0,2 \pi]$.