Suppose $M_1=(\mathbb{R}^2,g_s)$ is the plane with standard flat metric, it is a complete manifold. Now if I delete point origin, $M_2=(\mathbb{R}^2\setminus\{0\},g_s)$ is obviously not complete. However, when punctured plane is given a different metirc, it becomes complete, such as $$g=\frac{1}{|x|^2}g_s$$
I think it is geodesically complete, since when points get close to origin, the metric blows up. However, I don't know how to rigorously prove it?
Write $g_s$ in polar coordinates: $g_s=dr^2 + r^2\,d\theta^2$. Then, with the change of variables $s=\ln r$, it holds that $g=ds^2 + d\theta^2$. This is the standard metric on the cylinder $\mathbb{R}\times S^1$.