I am quite familiar with use of Wick rotations in QFT, but one thing annoys me: let's say we perform it for treating more conveniently (ie. making converge) a functional integral containing spinors; when we perform this Wick rotation, in a way we change the metric to (-,+,+,+) to (+,+,+,+), so the invariant group is no more SO(3,1) but SO(4) and (SO(4) being compact and the spinor representation non unitary) spinors don't carry finite dimensional representation of this group. So I feel like we shouldn't be talking anymore about this objects, but only about vectors of SO(4).

Is my fear justified? or where am I wrong in my reasoning?

I am quite familiar with use of Wick rotations in QFT, but one thing annoys me: let's say we perform it for treating more conveniently (ie. making converge) a functional integral containing spinors; when we perform this Wick rotation, in a way we change the metric to $(-,+,+,+)$ to $(+,+,+,+)$, so the invariant group is no more $SO(3,1)$ but $SO(4)$ and ($SO(4)$ being compact and the spinor representation non unitary) spinors don't carry finite dimensional representation of this group. So I feel like we shouldn't be talking anymore about this objects, but only about vectors of $SO(4)$.

Is my fear justified? or where am I wrong in my reasoning?