Let $\mathbb C$ be the complex plane, the lattice $\Lambda_1=\mathbb Z+i\mathbb Z$ is spanned by two vectors $1,i$, and $\Lambda_2=\mathbb Z+2i\mathbb Z$ by $1,2i$. Then tori $T_1=\mathbb C/\Lambda_1$ and $T_2=\mathbb C/\Lambda_2$ have different complex structures (since they correspond to different points in the fundamental domain), although they are diffeomorphic.
By lifting $T_1$ and $T_2$ to $\mathbb C$, there is a diffeomorphic map $$f:\mathbb C\to \mathbb C,$$ $$(z,\bar z)\mapsto (w,\bar w)$$ which sends $1$ to $1$ and $i$ to $2i$, where $z=x+iy$ and $w=u+iv$, then we get $x=u$ and $2y=v$, so $f$ can be written as $w=\frac{3}{2}z-\frac{1}{2}\bar z$, and the Jacobian matrix can be written as $$ J=\begin{pmatrix} \frac{\partial w}{\partial z}&\frac{\partial w}{\partial \bar z}\\ \frac{\partial \bar w}{\partial z}&\frac{\partial \bar w}{\partial \bar z} \end{pmatrix}=\begin{pmatrix} \frac{3}{2}&-\frac{1}{2}\\ -\frac{1}{2}&\frac{3}{2} \end{pmatrix}, $$ then according to Kodaira (see Morrow & Kodaira's book Complex manifolds p.151, Prop 1.2), the complex structure of $T_2$ can be represented by a Beltrami differentail $\varphi\in A^{0,1}(X,T^{1,0}_X)$, where $\varphi$ can be written as $=\varphi^i_{\bar j}d\bar z^j\otimes\frac{\partial}{\partial z^i}$ locally. And in our case $\varphi=\varphi^1_{\bar 1}d\bar z\otimes \frac{\partial}{\partial z}$, where $\varphi^1_{\bar 1}=\frac{\frac{\partial w}{\partial \bar z}}{\frac{\partial w}{\partial z}}=-\frac{1}{3}$ (see M&K p.150, line 8 for definition) which is not $0$, so $T_2$ and $T_1$ have different complex structures according to Kodaira's deformation theory.
What I feel puzzled is that: why diffeomorphism $f$ induces two different complex structures on the same real manifold underlying both $T_1$ and $T_2$, but does not induce different complex structures on $\mathbb C$ (see my previous question)?
My initial mativation is to understand the statement "the complex structure on $X_t$ is represented by $\varphi(t)$", so I represent the complex structure on $T_2$ relative to $T_1$ by $\varphi=-\frac{1}{3}d\bar z\frac{\partial}{\partial z}$, it is not zero, so they have different complex structures, till now, everything seems good? but when I represent the complex structures of $\mathbb C$ also by $\varphi$, some problems occur, we also have non-zero $\varphi=-\frac{1}{3}d\bar z\frac{\partial}{\partial z}$ for $\mathbb C$, it seems $f$ induces two differet complex structures on $\mathbb C$ as well, but it is not the case, I know I must have taken something wrong, but I don't know from which step, I made the mistake, can someone help me find the bug?
Well, it does. Your map $$f: \mathbb C \to \mathbb C, \quad x+iy \mapsto x + 2 iy$$ is not holomorphic with respect to the standard complex structure on $\mathbb C$.