Oops, there is an important omission in my above comment. I spent a lot of time talking about what a particle is and didn't say what a field is. Whereas particles are related to energy eigenstates, the field is related to eigenstates of the field value. These are two very different things and I think they are both valid, depending on what you are considering.

To me this is what it means to be both a field and a particle. At the same time, speaking of semantics, you can say particles don't exist because they are not a physical thing like a field value is.

Here is an aside that I just thought about while writing this which I hadn't considered before - Particles versus fields is similar to the Pauli Exclusion Principal. The Pauli Exclusion Principle basically states that energy/momentum can not be specified at the same time as position, and this is because they are eigenstates of different operators (energy/momentum operator and the time/position operator). You can't be in a pure state of both at the same time. In the particles versus field case, particles are related to eigenstates of the energy operator and fields are eigenstates of the field operator. These two values also can not be specified at that same time, for the same reason as in the Pauli Exclusion case.