I’m not sure, but it seems to me that the Zygote gradient is correct there. If you take the constant to be a + bi and expand the function, we get
f(x\equiv x_r + ix_i) = real[ (a + bi) (x_r + i x_i) ] = real[ax_r + i^2 (bx_i) + i(bx_r + ax_i)] = ax_r -bx_i
where the imaginary part of the constant, b, appears negative in the real part of the result. Thus the derivative of f relative to the imaginary part of x seems to be indeed -b.
I think this is the first time I differentiate a complex function, so take it with a grain of salt…