As electron is accelerated nearly to the speed of light what would occur? What would occur if an electron at rest was accelerated nearly to the speed of light? Would the photons emitted by the electron approach a gamma wave photon as it approached the speed of light, or since an electron is an excitation of the electron field would it just disappear into pure energy?
 A: 
What would occur if an electron at rest was accelerated to the speed of light? 

Any charged particle can be accelerated its speed increasing as more energy is supplied, but the limit of the speed is the speed of light.

At a Lorentz factor ( = particle energy/rest mass = [104.5 GeV/0.511 MeV]) of over 200,000, LEP still holds the particle accelerator speed record, extremely close to the limiting speed of light.

You:

Would the photons emitted by the electron approach a gamma wave photon as it approached the speed of light,

Part of the energy supplied to accelerate to high speeds will be radiated away, some of it might be in high energy photons, gamma rays, but there is a spectrum, mostly of lower energies. .
You ask

or since an electron is an excitation of the electric field would it just disappear into pure energy?

An electron is not an excitation of the electric field. You are confusing classical fields with quantum field theory, which posits a field for each elementary particle, electron field, photon field , neutrino field,... and the corresponding particle appears as an excitation of the corresponding field.
Also there is no "pure energy" in our models of physics. Energy is always an attribute of a particle or a field.
A: Electrons can be accelerated to the speed of light (or practically to the speed of light). If you accelerate electrons to merely 5 MeV the velocity of 0.996 c where c is velocity of light, and yes if they are accelerated to that velocity they will emit gamma like radiation. 
Here I would like to clarify that the term gamma radiation is mostly used for the electromagnetic radiation of nuclear origin. The super hard x rays (of wavelengths equal or lesser than gamma radiation) are still called x rays as they are of atomic origin. 
Particle accelerators especially electron accelerators (aka electron synchrotrons) are the world's brightest and most powerful x ray sources. They can generate the x rays of wavelengths as short as 0.01 angstroms. 
There are celestial bodies which accelerate matter to great velocities (energies) such as active galactic nuclei and emit much stronger and shorter x rays.
Regards,
