Problem of finding a very independent set of size at least $k$ is NP-Complete
Your notion is what is called distance-2 independent sets, also a particular case of rulling sets (see here Independent set where two vertices need to have distance &gt;= c). 

Your problem is NP-complete by an easy reduction to Independent Set (which is NP-complete, https://en.wikipedia.org/wiki/Independent_set_(graph_theory)#Maximum_independent_sets_and_maximum_cliques): for every vertex $u,v$ of $G$ such that there exists $w$ with $(u,w) \in E$ and $(v,w) \in E$, add the edge $(u,v)$.

This gives you a new graph $G'$ whose independent sets are the distance-2 independent sets of $G$.