Complexity of K-Colorful Coloring Problem for a Hypergraph
A strong k-coloring of a hypergraph assigns distinct colors to every member of a hyperedge and uses $k$ colors. When the hypergraph is $k$-uniform, this problem is equivalent to the problem you describe. Further, this problem is NPC as shown by Colbourn, Jungnickel and Rosa [1].

You can also prove yourself that this problem is NPC by a straightforward reduction from chromatic number. Essentially, you can add enough vertices around every edge and make them into cliques and you can make the reduction work.



[1] Colbourn, Charles J., Dieter Jungnickel, and Alexander Rosa. "The strong chromatic number of partial triple systems." Discrete applied mathematics 20.1 (1988): 31-38.