$\{N(t), t\geq 0\}$ is a Poisson process. If I classify all the odd events in the process as one type and even events into other type, I get two counting processes.
$N_i(t)$ be the number of type $i$ events that occur during $(0,t].$ $(i=1,2)$.
Am I correct in saying that I have used Bernoulli splitting of the given process with probabilities $\frac{1}{2}$ and $\frac{1}{2}$. And the new $N_i(t)$ are also Poisson processes and that they are independent of each other?
Let $\{T_n:n=1,2,\ldots\}$ be the arrival times of $\{N(t)\}$. If we define \begin{align} N_1(t) &= \sum_{n=1}^\infty \mathsf 1_{(0,t]}(T_{2n-1})\\ N_2(t) &= \sum_{n=1}^\infty \mathsf 1_{(0,t]}(T_{2n}), \end{align} then $\{T_{2(n+1)-1}-T_{2n-1}\}$ is an i.i.d. sequence of $\mathrm{Erlang}(2,\lambda)$ random variables, as is $\{T_{2(n+1)}-T_{2n}\}$. So $\{N_1(t)\}$ and $\{N_2(t)\}$ are renewal processes each with interrenewal density $$(\lambda t)\lambda e^{-\lambda t}\mathsf 1_{(0,\infty)}(t) $$ (where $\lambda>0$ is the intensity of $\{N(t)\}$) but are not independent, as $$\mathbb P(N_1(t)-N_2(t)\in\{0,1\})=1$$ for all $t\geqslant 0$.
Let $Z(t)=N_1(t)-N_2(t)$, then $\{Z(t):t\in\mathbb R_+\}$ is an alternating renewal process with uptime and downtime distributions both $\mathrm{Exp}(\lambda)$. That is pretty much all that can be said of this deterministically split Poisson process, I believe.