jmc

In the parallelogram, suppose that $d$ is opposite $a.$



Recall that in any parallelogram, the midpoints of the diagonals coincide. Hence, $$\frac{a+d}{2}=\frac{b+c}{2},$$which leads to $d=b+c-a.$

The complex number $d$ can also be opposite $b$ or $c,$ which leads to the possible values $d=a+c-b$ and $d=a+b-c.$



Thus, the possible values of $d$ are $$\begin{array}{rl}b+c-a& =(1-2i)+(8-5i)-(4+3i)=\boxed{5-10i},\\ a+c-b& =(4+3i)+(8-5i)-(1-2i)=\boxed{11},\\ a+b-c& =(4+3i)+(1-2i)-(8-5i)=\boxed{-3+6i}.\end{array}$$