Reading rule 03.04.02 on page 13 of 3rd Ed. Revised, the third paragraph reads:
"When a fighter or small craft launches, it ends movement for the impulse of launch in the same hex as its carrier, stacked on top of the carrier's counter but on any heading the launching player chooses."
So based on that you can be heading any direction you want when you launch whether or not the carrier (or base) is moving.