<s>[INST] Given $f(z)=\dfrac{U(z)}{V(z)}=\dfrac{2z^3-3z^2+7z-8}{z^4-5z^3+4z^2-6z+1}$ find $f(1-\sqrt{2}i)$ without lots of complex arithmetic. Such a problem is usually done either by direct substitution (ugh!) or synthetic division.
Synthetic division after several complex products and additions gives $U(1-\sqrt{2}i)=-8-3\sqrt{2}i$
After several more complex operations one finds, barring errors, that $V(1-\sqrt{2}i)=9+7\sqrt{2}i$ 
leaving the solution to the straightforward evaluation of
\begin{equation}
 f(1-\sqrt{2}i)=\frac{-8-3\sqrt{2}i}{9+7\sqrt{2}i}
\end{equation}
All standard stuff. The question is this: Can this be done without so many complex operations?
 [/INST] Since it has been 12 hours and no more answers have been suggested, I am posting a solution based upon a suggestion by @almagest.
\begin{equation}
 [z-(1-\sqrt{2}i)]\cdot[z-(1+\sqrt{2}i)]=z^2-2z+3
 \end{equation}
Using long division (or better yet, quadratic synthetic division) no complex arithmetic is required to obtain
\begin{equation}
 U(z)=(2z-1)(z^2-2z+3)+3z-11
 \end{equation}
and
\begin{equation}
 V(z)=(z^2-3z-5)(z^2-2z+3)-7z+16
 \end{equation}
Therefore
\begin{equation}
 U(1-\sqrt{2}i)=0+3(1-\sqrt{2}i)-11=-8-3\sqrt{2}i
 \end{equation}
and
\begin{equation}
V(1-\sqrt{2}i)=0-7(1-\sqrt{2}i)+16=9+7\sqrt{2}i
 \end{equation}
Therefore
\begin{align}
 f(1-\sqrt{2}i)&=\frac{-8-3\sqrt{2}i}{9+7\sqrt{2}i}\\
              &=\frac{-8-3\sqrt{2}i}{9+7\sqrt{2}i}\cdot\frac{9-7\sqrt{2}i}{9-7\sqrt{2}i}\\
              &=\frac{-114+29\sqrt{2}i}{179}
 \end{align}
</s>