Past exam of the mathematical methods I course of the University of Oxford / 2019 resit / 2

Here $i^3=-i$, so the argument condition concerns $z+i$. Writing $z=x+iy$, the complex number $x+i(y+1)$ must point along the direction $1+i$. ConsequentlyThis is an open ray in the Argand plane, rather than the whole straight line. Its endpoint $z=-i$ is excluded because the argument of zero is undefined; the opposite half-line has a different argument.