In this paper we study completely strongly prime elements (cs-prime elements) of a frame, defined by means of arbitrary joins as a strengthening of the classical notion of a prime element. We first show that, for a frame, this notion coincides with the previously considered notion of a completely prime (c-prime) element, so that no genuinely new class of elements is obtained; we also exhibit, by means of an explicit example, a (non-distributive) lattice in which this coincidence fails, showing that the frame hypothesis cannot be dropped. The cs-prime formulation is, however, a convenient working characterization, which we use throughout the paper together with the classical lattice-theoretic notions of completely join-prime and completely join-irreducible elements. Using this characterization we obtain several equivalent descriptions of (completely) prime elements and we investigate their behaviour under various frame-theoretic constructions, namely frame homomorphisms, quotient frames, ∧-factorization frames and product frames.