So... I managed to get what I needed and figured I'd answer my own question in case someone else has the same problem.
The Master Measure description is saved as Meta Data...
Once you have the list of master measures (using AppExtensions.GetMeasureList(app)), you can loop through to get the relevant info for each one. To get the meta data, use the below method:
item.Meta.Get<type>(meta data name)
The below example uses a class I created called MasterItem which simply holds the Type (Dimension or Measure), Title and Description.
List<MasterItem> mItems = new List<MasterItem>();
var mList = AppExtensions.GetMeasureList(app);
foreach (var item in mList.Items)
MasterItem m = new MasterItem();
m.Type = "Measure";
m.Title = item.Data.Title;
m.Description = item.Meta.Get<string>("description");