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I have a question about degrees of fridom.

Electron is 1/2 spin particle so it needs 2 component wave function. But instead haveing 1 equation of second order we linearize and have two equations of order 1 for two spinors and these two equations can be re-written in one equation for 4 component spinor.

But this four component wave function has complex components ...and so there are eight independent elements in wave function.

What I am confused is that electron has degrees of freedom 2 (spin up/down) and wave function 8 ? correcct or what I dont understand? or complexity of wave function does not account in degrees of freedom?

Thanks

Nick