What type of fields existed in early universe? In quantum field theory we associate field to every particle. So how many elementary fields exits in nature? Why are fields associated to particles different from fields associated to fundamental forces? And if fields are fundamental and existed since universe formed, why are we trying to develop other theories?
 A: Fields are a general concept in physics, aiding in describing forces and their interactions in classical physics, and interactions in quantum physics.

A field can be classified as a scalar field, a vector field, a spinor field or a tensor field according to whether the represented physical quantity is a scalar, a vector, a spinor, or a tensor, respectively. A field has a unique tensorial character in every point where it is defined: i.e. a field cannot be a scalar field somewhere and a vector field somewhere else. For example, the Newtonian gravitational field is a vector field: specifying its value at a point in spacetime requires three numbers, the components of the gravitational field vector at that point. Moreover, within each category (scalar, vector, tensor), a field can be either a classical field or a quantum field, depending on whether it is characterized by numbers or quantum operators respectively.

Quantum field theory is based on quantum mechanics. The standard model fields , each particle in the axiomatic table assumed to be a field, mathematically is represented by the free particle wave function of the corresponding equations (dirac for fermions, klein gordon for for spin0, maxwell quantized for photons .... . Creation and annihilation operators operating on these fields can be used to model   particle interactions. The Feynman diagrams are the tool for writing the integrals that have to be caluclated in order to get real numbers, crossection and decays ... 

So how many elementary fields exits in nature?

as many as the current standard model  table has. If/when this is modified by new data, there will be correspondingly more fields. Each extension of the standard model fills the universe ( hypothetically, it is just mathematical models) with new fields. 

why is field associated to particles different from field associated to fundamental forces?

The three fundamental forces of the standard model, are also characterized by particles and their fields, photon for electromagnetic, W and Z for weak, gluon for strong. When/if gravity is quantized a new model becomes standard there is also the graviton.

And if fields are fundamental that they existed since universe formed. 

The current standard model for the universe, the Big Bang , has an inflaton field in the beginning of it.

why are we trying for other theories? is Quantum field theory not correct or complete? 

Quantum field theory is a mathematical tool used in all models. It is not a theory. The models are the theory, and these still are changing because there are open questions in the present standard models both for the particles and the universe.
A: The fundamental fields of the Standard Model are the various quarks (up, down, ..), leptons (electron, electron neutrino, mu, ..), gauge bosons and the Higgs boson, where the matter fields differ from the force fields by their statistics - gauge bosons are, well, bosonic, like the Higgs field, whereas matter fields are fermionic. It is not plausible that these fields existed literally 'since the universe formed': QFT is not complete as it does not include gravity, and it seems to be impossible to extend it in a naive way to make it do so. 
Hence, it seems that the Standard Model fields should arise as some low-energy limit to a more fundamental theory of everything, which does include gravity. Which theory this should be is a matter of active research, although many think it is probably string theory.
