"Completion Design" is a big topic... what exactly do you mean? If you're starting from scratch, then you're looking at the lower completion: perforated? bare foot? sand control? Then on to inflow/ outflow curves to decide the tubing size, and then deciding if stimulation or artificial lift is needed. Some companies call the guys that do the completion design up to this concept stage "Production Technologists".
After the concept has been selected (often for the entire field), the particular well completion has to be designed. The first thing is the stress analysis: what tubing weight do you need, or could you use things like PBRs, or land the tubing in tension to reduce the in service stress. Then you work out the the exact depths of stuff, the amount of tubing, and any jewellry: SSSVs, nipples, PBRs. Then you can write your completion programme, knowing volumes, depths, pressures etc working out what fluids etc you're using, often going back to the Production Technology or Reservoir people for guidance on formation damage considerations and stuff.