What does happen if the ratio of an ideal transformer is complex?
In a normal ideal transformer you have:
\$ V_2=m V_1 \$ and \$ I_1=mI_2\$.

where m is the transformer ratio.

In this special ideal transformer you'll have just the same, except that all variables are phasors (shown in bold) and asterisk symbol is the complex conjugate operation:

\${\bf V_2}= {\bf m} {\bf V_1}\$ and \${\bf I_1}={\bf m}^*{\bf I_2} \$ 

If you apply those equations there will be no change in power between primary and secondary, and the two-ports is reciprocal.