An ordinal $\alpha$ is called +1 stable, if $L_\alpha <_1 L_{\alpha+1}$

By considering $Σ_n$ elementary submodel we can generalize it.

I'm curious about its further generalizations.

Does there exist an ordinal $\alpha$, such that $L_{\alpha+1}$ is elementary equivalent to $L_{\beta+1}$ for some larger $\beta$ where every element of $L_\alpha$ is added in the language? 

Can we change "+1" to "+n", or "+α"? Is it possible that $L_{\alpha2} \cong L_{\beta 2}$ for the language $\{\in,L_\alpha\}$?