Completeness in same space but two metrics

Completeness in same space but two metrics

Let $(X,\rho)$ and $(X,\sigma)$ be metric spaces. If $X$ is
$\rho$-complete is $X$ $\sigma$-complete? Justify your answer.
A little bit of reference showed that this is not necessarily the case. So
I have to give a counter example of a space which is $\rho$-complete but
not $\sigma$-complete.
Any hints/ideas? Thanks in advance.