This is a Squammipinnis anthias, usually referred to as a Lyretail anthias. Usually these fish are female, but one within a harem will turn male. Visually, it will change from orange to purple and exhibit dominance over the others. Lyretails are usually best kept in an odd numbered group, such as five or seven.
Lyretail males are bright purple, while the females (as pictured in the lower right) are bright orange with what looks like vivid blue eye shadow. I've been keeping Lyretails in my reef since 2008. It's important that these fish are fed often, and an automatic feeder is just the tool to assist a hobbyist to fill this need. Due to their activity, respiration is greater and during acclimation I highly recommend getting them into the aquarium in under 40 minutes, or add an airstone to provide sufficient oxygenation. I'd definitely recommend putting them in a quarantine tank to make sure they aren't harboring any diseases before introducing them to your reef tank.