No, the worker bees do not kill each other in the process of becoming the queen. When multiple new queens emerge from their queen cells, they typically engage in fights with each other. The queens will face off, and the fights involve stinging and potentially killing each other; however, it is the queens doing the fighting, not the worker bees. This fighting among the queens is a natural process whereby the strongest or most dominant queen survives, becoming the new reproductive leader of the colony.
Once a new queen has emerged, she will begin her reign by mating with drones and then starting to lay eggs. The worker bees support the new queen and will often eliminate any remaining rival queens that did not win the fight to assume control. This process ensures that the colony remains led by a single, healthy queen, maintaining its organization and productivity.