$su(1,1) \cong su(2)$?

The ladder operators do belong to the real Lie algebras$^1$ $$\begin{align} su(1,1)~:=~&\{m\in {\rm Mat}_{2\times 2}(\mathbb{C}) \mid m^{\dagger}\sigma_3=-\sigma_3m,~ {\rm tr}(m)=0\}\cr ~=~&{\rm span}_{\mathbb{R}}\{ \sigma_1, \sigma_2, i\sigma_3 \}\cr ~\cong~sl(2,\mathbb{R}) ~:=~&\{m\in {\rm Mat}_{2\times 2}(\mathbb{R}) \mid {\rm tr}(m)=0\}\cr ~=~&{\rm span}_{\mathbb{R}}\{ \sigma_1, i\sigma_2, \sigma_3 \}\cr ~=~&{\rm span}_{\mathbb{R}}\{ \sigma_+, \sigma_-, \sigma_3 \}\cr ~\cong~ so(2,1)~:=~&\{m\in {\rm Mat}_{3\times 3}(\mathbb{R}) \mid m^{t}\eta =- \eta m\}, \end{align}$$ $$ \sigma_{\pm}~:=~\frac{\sigma_1\pm i \sigma_2}{2}, \qquad \eta~=~{\rm diag}(1,1,-1), $$ but they do not belong to the real Lie algebras $$\begin{align} su(2)~:=~&\{m\in {\rm Mat}_{2\times 2}(\mathbb{C}) \mid m^{\dagger}=-m,~ {\rm tr}(m)=0\}\cr ~=~&{\rm span}_{\mathbb{R}}\{ i\sigma_1, i\sigma_2, i\sigma_3 \} \cr ~\cong~ so(3)~:=~&\{m\in {\rm Mat}_{3\times 3}(\mathbb{R}) \mid m^{t}=-m\}. \end{align}$$ All the above 5 real Lie algebras have complexifications isomorphic to $$\begin{align}sl(2,\mathbb{C})~:=~&\{m\in {\rm Mat}_{2\times 2}(\mathbb{C}) \mid {\rm tr}(m)=0\}\cr ~=~&{\rm span}_{\mathbb{C}}\{ \sigma_1, \sigma_2, \sigma_3 \}.\end{align}$$

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$^1$ Here we follow the mathematical definition of a real Lie algebra. Be aware that in much of the physics literature, the definition of a real Lie algebra is multiplied with a conventional extra factor of the imaginary unit $i$, cf. footnote 1 in my Phys.SE answer here.

You may indeed identify the generators in the way you did. However, the Lie algebras and Lie groups are different because – as quickly said by Qmechanic – you must use different reality conditions for the coefficients.

A general matrix in the $SU(2)$ group is written as $$ M = \exp[ i( \alpha J_+ + \bar\alpha J_- + \gamma J_0 )] $$ where $\alpha\in {\mathbb C}$ and $\gamma\in {\mathbb R}$ while the general matrix in $SU(1,1)$ is given by $$ M' = \exp [ i( \alpha_+ J_+ + \alpha_- J_- + \beta J_0 ] $$ where $\alpha_+,\alpha_-,\beta\in {\mathbb R}$ are three different real numbers.

To summarize, for $SU(2)$, the coefficients in front of $J_\pm$ are complex numbers conjugate to each other, while for $SU(1,1)$, they are two independent real numbers. (And I apologize that I am not sure whether the $i$ should be omitted in the exponent of $SU(1,1)$ only according to your convention. Probably.)

If you allow all three coefficients in front of $J_\pm,J_0$ to be three independent complex numbers, you will obtain the complexification of the group. And as Qmechanic also wrote, the complexification of both $SU(2)$ and $SU(1,1)$ is indeed the same, namely $SL(2,{\mathbb C})$.