Infinitely many imaginary quadratic fields in which p splits completely?
The Dedekind-Kummer theorem tells us in quadratic number fields $\mathbb Q(\sqrt d)$, $p$ splits iff $\left(\frac \Delta p \right) = 1$, that is using Kronecker symbol, $\Delta$ is a quadratic residue for odd $p$ or $\pm 1 \pmod 8$ for $p = 2$.
Recall $\Delta$ is the discriminant, which is $4d$ for $d \equiv 2,3 \pmod 4$ or $d$ for $d \equiv 1 \pmod 4$. Then it should be easy to reason about the $d$ such that $p$ splits.