  A $4$-general set in ${\rm PG}(n,q)$ is a set of points of ${\rm PG}(n,q)$
spanning the whole ${\rm PG}(n,q)$ and such that no four of them are on a
plane. Such a pointset is said to be complete if it is not contained in a
larger $4$-general set of ${\rm PG}(n, q)$. In this paper upper and lower
bounds for the size of the largest and the smallest complete $4$-general set in
${\rm PG}(n,q)$, respectively, are investigated. Complete $4$-general sets in
${\rm PG}(n,q)$, $q \in \{3,4\}$, whose size is close to the theoretical upper
bound are provided. Further results are also presented, including a description
of the complete $4$-general sets in projective spaces of small dimension over
small fields and the construction of a transitive $4$-general set of size $3(q
+ 1)$ in ${\rm PG}(5, q)$, $q \equiv 1 \pmod{3}$.
