Patrick Hamill, actually, borrowed that problem from Goldstein (see- $3^{rd}$ ed. Chapter 9, derivations 1), and unfortunately, misphrased it. So I'm just quoting the problem from Goldstein, and my interpretation of it, and its solution for your part of that problem with a personal advice.
One of the attempts at combining the two sets of Hamilton’s equations
into one tries to take q and p as forming a complex quantity. Show
directly from Hamilton’s equations of motion that for a system of one
degree of freedom the transformation $$Q=q+ip,$$ $$P=Q^*$$ is not
canonical if the Hamiltonian is left unaltered. Can you find another
set of coordinates $Q'$, $P'$ that are related to $Q$,$P$ by a change
of scale only, are they canonical?
My interpretation of the unaltered Hamiltonian: Given a Hamiltonian, and if you perform this transformation the form of the Hamiltonian will remain the same that is $H(Q,P)=H(q,p)$. As the transformation has no time dependent term, so no PD of generating function wrt time is taken into account.
Now applying inverse transformation $$q={1 \over 2}(Q+P),$$ $$p={1 \over 2i}(Q-P)$$ And we expect $$\dot Q= \dot q +i\dot p.$$ Now from Hamilton's equation of motion $$\dot Q= {\partial H \over \partial P}={\partial H \over \partial q}{\partial q \over \partial P}+{\partial H \over \partial p}{\partial p \over \partial P}={- \dot p . {1 \over 2}}+ {{-1 \over 2i} \dot q}={{-1 \over 2} (\dot p-i \dot q)}$$ which is obviously not what we expected and this proves the noncanonicity. The condition part of your question plays its role when we consider the second part of Goldstein's problem whose solution is given in Homer Reed.
Please don't use Hamill for self study. It's full of error. If there's anything useful in that book, that's its bibliography. Choose a mainstream textbook mentioned there.
