Question

what name has been given to those organisms which breakdown the complex organic compounds present in dead animals and plants ?

Answer 1

these are microbes /decomposers/ microorganisms
by saprophytic mode of nutrition

Answer 2

The plant which derive their food from dead and decaying organic matter are called saprophytes or saprophytic plants of saprotrophs.
Saprotrophs secrete digestive juices on the dead and decaying matter and convert them into solution. Then they absorb the nutrients from it.