So taking your suggestion to use a quadratic number field,  I get rid
of syntax errors at last.   But I guess the problem is too difficult
as no answer comes back in a few minutes.
I let Mathematica run a similar problem for 36 hours with no reply;
but I don't understand why it's too difficult.  Seems like it should
be 16 polynomial quasi-divisions.   I guess the
coefficient expressions must blow up.   But even if they have
thousands of terms,  polynomial division is quadratic so should happen
in a few seconds.

sage: R.<i> = QQ[i]
sage: K.<z,p,d,e,f,m,g,h,l,r> = R[]
sage: F = i*m*z^14 + (d+g)*z^13 + i*(f-p)*z^12 + e*z^11 + i*f*z^10 +
(e-g)*z^9 + i*(m+p)*z^8 + 2*d*z^7 - i*(m+p)*z^6 + (e-g)*z^5 -i*f*z^4 +
e*z^3 - i*(f-p)*z^2 + (d+g)*z - i*m
sage: G = d*z^16 - (l+i*p)*z^15 + (e+i*h)*z^14 - r*z^13 + e*z^12 - (r
- i*p)*z^11 + (d-i*h)*z^10 - l*z^9 + l*z^7 - (d + i*h)*z^6 + (r+i*p)
*z^5 - e*z^4 + r*z^3 - (e-i*h)*z^2 + (l-i*p)*z -d
sage: F.resultant(G)

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