Counting process which is not a Poisson process

Let $\{N_t, t\geq 0\}$ be a homogeneous Poisson process with intensity $\lambda$, it holds that $N_{t+s}-N_s\sim Pois(\lambda t)$. Thus, let $Z_t=N_{t+s}-N_{s}$ for any $t,s\geq 0$. Then clearly $\{Z_t, t\geq 0\}$ is a counting process, $Z_t\sim Pois(\lambda t)$, but most importantly - its increments are dependent, e.g. let $m>s\geq 0,\,t_2>t_1\geq 0$ and then $Z_{t_2}-Z_{t_1}=N_{t_2+s}-N_{t_1+s}$ and $Z_{t_1}-Z_{0}=N_{t_1+m}-N_{m}$, intervals $\left(t_1,t_2\right]$ and $(0,t_1]$ do not overlap, but $N_{t_2+s}-N_{t_1+s}$ and $N_{t_1+m}-N_{m}$ are dependent since $m>s$.