Fermions are not necessarily complex. Majorana spinors fulfill a reality condition $\Psi=\Psi^\mathcal{C}$, and in the Majorana basis a Majorana spinor is manifestly real. Clearly, only neutral fermions can be described by Majorana spinors, and neutrinos are candidates for such fermions, even though at present it is not clear whether they are Dirac or Majorana particles.

Fermions are not necessarily complex. Majorana spinors fulfill a reality condition $\Psi=\Psi^\mathcal{C}$, and in the Majorana basis a Majorana spinor is manifestly real. Clearly, only neutral fermions can be described by Majorana spinors, and neutrinos are candidates for such fermions, even though at present it is not clear whether they are Dirac or Majorana particles.

I think that there are some issues with your question. The answer above is on the title "Why are fermions complex". In your question, you suggest that all fermions need to be described by the Dirac equation, which is not the case. There is also the Weyl equation, and some refer to the field equation of Majorana spinors as Majorana equation. And I cannot follow the statement on the degrees of freedom vs. structure either, photons are described by a vector and do not have a "structure" either, at least according to our present knowledge.