The Hermitian Veronesean in $PG(3,q^2)$, given by $\mathcal{V}:=\{ (1,x,x^q,x^{q+1}):x\in\mathbb{F}_q\}\cup\{(0,0,0,1)\}$, is a well-studied rational curve, and forms a {\em special} set of the Hermitian surface $H(3,q^2)$. In this paper, we give two local characterisations of the Hermitian Veronesean, based on sublines and triples of points in perspective.