Quantum fields in position space. Could it be considered as one simple harmonic oscillator at every point of space? Quantum fields in almost every note that a have seen are considered in momentum space. I visualize this as one harmonic oscillator at every point momentum space. then Quantum field in position space is rather a Fourier transform than some thing for its own. could we -as suggested by duality of momentum and quantum- start from position space and consider QF as one simple harmonic oscillator at every point of space? 
 A: 1) Yes, except that the oscillators are coupled. This  is the whole point of going to momentum space. For a free field $\phi(x)$, it's fourier transform $\tilde{\phi}(p)$ obeys the equation for SHM for each $p$, independent of what happens at $p+dp$. In words, at each point in momentum space, there is a harmonic oscillator evolving independently in time.
2) This is often written as "the degrees of freedom have decoupled in the momentum space. In position space, for a free field, the derivative term couples neighbouring degrees of freedom $\phi(x)$ and $\phi(x+dx)$. This coupling vanishes in the momentum space-$\tilde{\phi}(p)$ and $\tilde{\phi}(p+dp)$ evolve independently.
3) If this seems weird, then note that $\phi(x)$ is indeed a superposition of all those independent oscillators of momentum space. This is just what the fourier transform means. Now, the uncoupled oscillators in the  momentum space turn into free waves in the position space(i.e. superposition allowed and such nice properties). This is as expected intuitively-since the oscillators themselves are independent, adding another set of them does nothing-there are now 2 independent uncoupled sets of oscillators. The waves have 'superposed' and are still free waves.
