As a simple example, consider the complex-valued function $S[x]=x^2$, where $x\in\mathbb{C}$. The critical point corresponds to $\partial_x S=0$ which has the solution $x=0$. Now consider the quantity ${\rm Re}[iS[x]]$, which can be easily computed to be, $${\rm Re}[iS[x]]={\rm Re}\left[i\left({\rm Re}[x]+i{\rm Im}[x]\right)^2\right]={\rm Re}\left[-2{\rm Re}[x]{\rm Im}[x]+i\left({\rm Re}[x]^2-{\rm Im}[x]^2\right)\right]=-2{\rm Re}[x]{\rm Im}[x].$$ Now, I have to find the contour, which passes through the critical point, and along which ${\rm Re}[iS[x]]$ decreases or increases most rapidly. In the paper (section II) the answer is given to be along ${\rm Im}[x]=+{\rm Re}[x]$ is the steepest descent contour and ${\rm Im}[x]=-{\rm Re}[x]$ is the steepest ascent contour. But how do we get these solutions? What conditions/equations in general allows us to make such conclusions? For, example we were to work with a different function $S[x]=x^3$, the quantity ${\rm Re}[iS[x]]$ reads $${\rm Re}[iS[x]]={\rm Im}[x]^3-3{\rm Im}[x]{\rm Re}[x]^2.$$ Then what equations we must solve to get the steepest descent or ascent contours?
Edit: The steepest descent/ascent contours are supposed to pass through the critical points.