Do I need complex endorsement in each class or type of aircraft?
You only need one complex endorsement under FAA regulations, 


  §61.31   Type rating requirements, additional training, and
  authorization requirements.
  
  (c) Aircraft category, class, and type ratings: Limitations on the
  carriage of persons, or operating for compensation or hire. Unless a
  person holds a category, class, and type rating (if a class and type
  rating is required) that applies to the aircraft, that person may not
  act as pilot in command of an aircraft that is carrying another
  person, or is operated for compensation or hire. That person also may
  not act as pilot in command of that aircraft for compensation or hire.
  
  (e) Additional training required for operating complex airplanes. (1)
  Except as provided in paragraph (e)(2) of this section, no person may
  act as pilot in command of a complex airplane, unless the person has—
  
  (i) Received and logged ground and flight training from an authorized
  instructor in a complex airplane, or in a flight simulator or flight
  training device that is representative of a complex airplane, and has
  been found proficient in the operation and systems of the airplane;
  and
  
  (ii) Received a one-time endorsement in the pilot's logbook from an
  authorized instructor who certifies the person is proficient to
  operate a complex airplane.


In your case a class rating is only required for the "multi" aspect of the DA-42 as the DA-42 has no type rating. 

The FAA further more defines complex aircraft as 


  Complex airplane means an airplane that has a retractable landing
  gear, flaps, and a controllable pitch propeller, including airplanes
  equipped with an engine control system consisting of a digital
  computer and associated accessories for controlling the engine and
  propeller, such as a full authority digital engine control; or, in the
  case of a seaplane, flaps and a controllable pitch propeller,
  including seaplanes equipped with an engine control system consisting
  of a digital computer and associated accessories for controlling the
  engine and propeller, such as a full authority digital engine control.


No part of this definition stipulates how many engines the aircraft has.