The Level(3) name was much more memorable in the carrier/enterprise/ISP field and it feels like they are starting from scratch on brand recognition, because as you said, lots of people haven't heard of Centurylink before.
Edit: And as noted by the thread "CenturyLink is totally shady.", Centurylink's retail consumer reputation is quite tarnished (as are most large consumer ISP companies) and mixing that reputation with Level(3) was a bad decision, IMO.