The complex form of Hamilton canonical equations I found an excerpt on page 171 of "The variational principles of mechanics" written by Cornelius Lanczos stated that 

If, however, the conjugate variables $q_k$, $p_k$ are replaced by the complex variables  $$\frac{q_k+ip_k}{\sqrt{2}}=u_k$$ and $$\frac{q_k-ip_k}{\sqrt{2}}=u_k^*,$$ then the double set of canonical equations $$ \dot q_i=\frac{\partial H}{\partial p_i}, \qquad\dot p_i=-\frac{\partial H}{\partial q_i}$$ can be replace by the following single set of complex equations $$\frac{du_k}{idt}=-\frac{d H}{d u_k^*}.$$

I did not understand that why shall we introduce the complex conjugate of $u_k$ and the coefficient $\sqrt 2$. It seems to me that the equation $$\frac{du_k}{idt}=-\frac{d H}{d u_k}$$ with $$u_k=q_k+ip_k$$ can already fulfill the task. Could you please tell me why my thinking is wrong?
P.S. I found another link (Relationship between complex analysis and Hamilton's canonical equation) related to this problem but not exactly the same.
 A: *

*We need the complex conjugated variable$^1$ $\bar{z}$ because the Hamiltonian $H(z,\bar{z})$ is typically not holomorphic in $z$.


*The normalization of the CCR$^2$
$$\{z,\bar{z}\} ~=~-i \tag{1}$$
explains the $\sqrt{2}$-normalization in
$$ z~=~\frac{q+ip}{\sqrt{2}}.\tag{2}$$


*Recall that Hamilton's equations can be written in terms of the Poisson bracket
$$\dot{z}~=~\{ z, H\}~=~\{ z, z\}\partial H + \{ z, \bar{z}\}\bar{\partial} H. \tag{3}$$
In the last equality in eq. (3) we used that the Poisson bracket $\{ \cdot, \cdot\}$ is a first-order differential operator wrt. its second entry.


*We can now derive the complex form of Hamilton's equations
$$\dot{z} ~\stackrel{(1)+(3)}{=}~-i\bar{\partial}H.\tag{4}$$
--
$^1$ To connect to OP's notation, use $z\equiv u$.
$^2$ Quantum mechanically, the Poisson bracket (1) should be replaced with $(i\hbar)^{-1}$ times the corresponding commutator
$$[\hat{z},\hat{z}^{\dagger}] ~=~\hbar \hat{\mathbb{1}}, \tag{5}$$
in accordance with the correspondence principle between classical and quantum physics. Here the operator $\hat{z}$ and the Hermitian adjoint operator $\hat{z}^{\dagger}$ are an annihilation and a creation operator, respectively. Hamilton's equations (3) are then replaced by Heisenberg's equations of motion
$$i\hbar\dot{\hat{z}}~=~[ \hat{z}, \hat{H}].\tag{6} $$
