Using 1904 date system like in Excel

See my response here:

http://community.powerbi.com/t5/Desktop/Convert-json-DateTime-format/m-p/28712

Let me provide you a full answer and not just links and hints

You can achieve the conversion this way in Power Query

The step to execute to do the transformation

= Table.TransformColumns(#"<previous step name>",{{"<column name to update>", DateFromJson}})

The code of the Power Query custom function

let DateFromJson = (date as any) as any =>
     let
        input = if date is null then "/Date(00000000000000)/"else date,
         Stripped = if Text.StartsWith(input, "/Date(") and Text.EndsWith(input, ")/") then Text.Range(input, 6, Text.Length(input) - 8) else error "Not a date",
         Position = Text.PositionOfAny(Stripped, {"+", "-"}, 1),
         Parts = if Position < 0 then { Stripped, "0" } else { Text.Range(Stripped, 0, Position), Text.Range(Stripped, Position) },
         NumberParts = { Number.FromText(Parts{0}), Number.FromText(Parts{1}) },
         Result = Date.FromText("1/1/1970") + #duration(0, 0, 0, (NumberParts{0} + 36000 * NumberParts{1}) / 1000),
        output = if Date.Year(Result) = 1970 then null else Result

     in
  output 
 in DateFromJson