Electrons and positrons sort of share the same field, but not really. The complexity is that you cannot just point to a spot in space and say "the electron field here is energized." Electrons and positrons are spin-1/2 fermions, and QFT textbooks normally represent them with 4-component Dirac spinors. If you are not comfortable with the word "spinor" that is fine it is not imortant: the important term in this context is "4-component." The heuristic justification that I normally hear is that you get one component each for left-handed electrons, right-handed electrons, left-handed positrons, and right-handed positrons. Two spin states, direct product with two charge states. But, there is an extra tricky subtlety in that I cannot point to the 2nd component of the spinor and say "this is the size of the excitation of the right-handed electron field." In the Weyl basis that might actually be possible in the non-interacting relativistic limit, I am not sure, but in general each component could be a mixture of all four states.
The main take-away is that positrons are not just negative excitations of the electron field. Positrons have their own degrees of freedom that can fluctuate independently from the electron degrees of freedom. Charge conservation and spin conservation couple the positron and electron degrees of freedom in important ways, but a priori they are independent fields.
Most of this comes from bits of conversation, but Peskin and Schroeder page 45 might be a good place to go for more information.