This might be a silly question, but how did they get $15-15i$ for example 1? The operation they did didn't follow any of the definitions listed (a-d). Did they use operation a? If so, where did they get $z$ from because the question only gave $x$ & $y$ within $C^2$, and why did they use the conjugate of $y$?
Lastly, why does $\langle x,y\rangle=\langle y.x\rangle$ in definition c.
Expanding the product and using $i^2=-1$,
$$(1+i)(2+3i)+4(4-5i)=2+3i+2i+3i^2+16-20i$$
$$=2+5i-3+16-20i=15-15i.$$