On a discrete version of length metrics

Let $(X,d)$ be a metric space. We study a metric $d_0$ on $X$ naturally derived from $d$. If $(X,d)$ is complete and locally compact, or if it is complete and $(d_0)_0=d_0$, then $d_0$ coincides with the length metric induced by $d$. Counterexamples are constructed when any of the hypotheses is absent. The behavior of the iterates of $d_0$ (the metrics $d_0^n$ inductively defined as $(d_0^{n-1})_0$) is also considered.