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What I seek here is to understand whether the electric field in its pure form as in between the electron and the proton is uniform or does it have some kind of wave/particle nature or both, does it have frequency or wavelength, or is it quantized?.

I do not want any explanations of the effects of electric field or thereof, I want to understand the field itself. Please tell me something I all ready do not know from my engineering education.

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The electric field itself is not accessible by experiments. We can only observe e.g. trajectories of charged particle, etc., to find the forces they are subjected to. It all comes down to the electric field just being a theoretical concept used to describe the phenomena covered by electrodynamics.

Thus, we cannot make a definite statement on the nature of the electric field. It really depends on the theory you are considering. In classical electrodynamics as developed in the 19th century, the electric and magnetic fields are vector fields permeating all of space. I.e. for each position in space there is a corresponding electric and magnetic field vector, respectively. Of course, one can have a completely equivalent description in momentum (or Fourier) space such that one assigns an electric and magnetic field amplitude vector to each wave vector $\vec k$. The description in momentum space is more suited to describe the propagation of waves, e.g. dipole radiation, while the position space representation is used predominantly in electro-/magnetostatics.

In the early 20th century, special relativity has been developed and it became apparent that the electric and magnetic field are basically the same phenomenon and mix when changing the frame of reference. E.g. person A travels on a train and carries an electric charge. She will only observe (the effects of) an electric field. At the same time person B stands on the platform while the train passes. To her, the moving charge person A carries is basically a current flowing, i.e. she will observe (the effects of) a magnetic field. The Maxwell equations (describing classical electrodynamics) which usually are stated in terms of electric and magnetic fields can be recast in a ("covariant") form which is independent of the frame of reference. In this version, an electromagnetic field tensor appears in the equations which combines electric and magnetic field. Thus, in the relativistic description, a 4x4-component tensor is assigned to each point in four-dimensional space-time. Still, the theory is the same as the original version of classical electrodynamics but the electromagnetic field tensor is a very different object.

In the same year Einstein published his works on special relativity, he also published an explanation of the photoelectric effect which pointed towards light (EM-radiation) being constituted by photons, i.e. quantized massless particles carrying energy and momentum. There was also some other experimental evidence connected to black-body radiation. Finally, quantum mechanics was developed which has been developed even further to quantum field theory (specifically quantum electrodynamics, QED) where we again have a description in terms of fields, i.e. we assign an object to each point in space (position or momentum space). Within QED, these objects are operators for particle creation and annihilation. If you'd like an introduction to QED, I suggest R. P. Feynman's book "QED: The strange theory of light" which is written for the general public.

This overview of the evolution of our understanding of "light" is by no means complete. My point is that we cannot talk about the nature of the electric field. It's just not real, it's a theoretical construct for thinking in terms of classical, non-relativistic electrodynamics. As physics refined its understanding and new theories emerged, the notion of the electric field became overruled by other concepts which will in turn be overruled as science progresses further.

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However the question may be restated in terms of near-field part and far-field part of the EM field. These concepts do not depend on the reference frame, because, for instance only the far-field part transports energy-momentum. Is it possible to describe the near-field part in terms of photons? I do not think so, at least directly. –  V. Moretti Jun 15 at 12:21

The electric field is the continuum limit of the quantized photon field, as far as the standard model is concerned. Thus, in "reality" (as far as we consider the standard model reality and not itself an effective field theory) variations of measurement of the electric field are actually photons hitting your measurement device.

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It is not so simple: Photons have transversal degrees of freedom only as they are associated with the far-field part of EM field (decreasing as $1/r$ from the source). Electrostatic field or the near-field part including temporal and longitudinal degrees of freedom (decreasing as $1/r^2$ from the source). A continuous limit (coherent states if I understand well) would be enough to describe the electric field in terms of photons. –  V. Moretti Jun 15 at 12:15

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