I have two very related questions. Suppose $(M,J)$ is an almost complex manifold with complex structure $J$. Then for a $\varphi\colon M\to \mathbb{R}$ we define a two form: $$\omega_\varphi := -dd^\mathbb{C}\varphi,$$ where $d^\mathbb{C}\varphi(X) = d\varphi(J(X))$. My first question actually might be me making a silly stupid mistake, but why is $\omega_\varphi$ not automatically equal to $0$ seeing as we have somewhat of a $d^2$ if we combine the definitions (I know, $d\varphi$ is a 1-form, but precomposing it with $J$ somehow makes a difference, where $-dd\varphi(J(X))\neq 0$?).
My next question is assistance with using this definition to find $\omega_\varphi$ where $\varphi(z) = ||z||^2 = \sum z_i\bar{z}_i$. My main issue with doing this is I am not sure how to evaluate $d^\mathbb{C}\varphi$, whether this ends up being the directional derivative of $\varphi$ in the direction of a vector field, or what exactly it looks like.
Any hints are well appreciated, thanks!
For the relation to $d^2=0$, it is easier to work on complex manifolds in a complexified picture. In this case you get a decomposition into $(p,q)$-types on complex valued forms and correspondingly $d=\partial+\overline{\partial}$ where the two components raise the $p$-degree respectively the $q$-degree. Then $d^2=\partial^2+(\partial\overline{\partial}+\overline{\partial}\partial)+\overline{\partial}^2$, and for reasons of degree the three summands have to vanish separately. However, this does not imply that $\partial\overline{\partial}$ vanishes (and indeed it is highly non-trivial in general). The relation to the real $d$ and $d^{\mathbb C}$ you are looking roughly is that $\partial=\frac12(d+id^{\mathbb C})$ and $\overline{\partial}=\frac12(d-id^{\mathbb C})$. Alternatively, you can compute directly that $dd^{\mathbb C}\phi$ maps vector fields $\xi$ and $\eta$ to $\xi\cdot J(\eta)\cdot\phi-\eta\cdot J(\xi)\cdot\phi-J([\xi,\eta])\cdot\phi$, and there is no reason why this should vanish.
Hint for your second question: Write $z_j=x_j+iy_j$, so $\phi=\sum_j(x_j^2+y_j^2)$. Then compute using that $dx_j\circ J=-dy_j$ and $dy_j\circ J=dx_j$.