\documentclass{article}

\usepackage{agda}

\usepackage{amsmath}
\usepackage{newunicodechar}
\newunicodechar{₁}{\ensuremath{{}_{\text{1}}}}

\pagestyle{empty}

\begin{document}

\noindent Control:
\begin{code}
  _ : Set₁
  _ = Set
\end{code}

\noindent No extra indentation (alignment):
\begin{code}[hide]
  _ : Set₁
  _ =
\end{code}
\begin{code}
    Set
\end{code}

\noindent No extra indentation (indentation):
\begin{code}[hide]
  _ : Set₁
  _ =
\end{code}
\begin{code}
     Set
\end{code}

\noindent No extra indentation (alignment):
\begin{code}[hide]
  postulate
    _ :
\end{code}
\begin{code}
      Set
\end{code}

\noindent No extra indentation (indentation):
\begin{code}[hide]
  postulate
    _ :
\end{code}
\begin{code}
       Set
\end{code}

\noindent No extra indentation (alignment, \texttt{AgdaMultiCode}):
\begin{AgdaMultiCode}
\begin{code}[hide]
  _ : Set₁
  _ =
\end{code}
\begin{code}
    Set
\end{code}
\begin{code}[hide]
  _ : Set₁
  _ =
\end{code}
\begin{code}
    Set
\end{code}
\end{AgdaMultiCode}

\noindent Extra indentation (indentation, \texttt{AgdaMultiCode}):
\begin{AgdaMultiCode}
\begin{code}[hide]
  _ : Set₁
  _ =
\end{code}
\begin{code}
     Set
\end{code}
\begin{code}[hide]
  _ : Set₁
  _ =
\end{code}
\begin{code}
     Set
\end{code}
\end{AgdaMultiCode}

\noindent No extra indentation (alignment, \texttt{AgdaMultiCode}):
\begin{AgdaMultiCode}
\begin{code}[hide]
  postulate
    _ :
\end{code}
\begin{code}
      Set
\end{code}
\begin{code}[hide]
  postulate
    _ :
\end{code}
\begin{code}
      Set
\end{code}
\end{AgdaMultiCode}

\noindent Extra indentation (indentation, \texttt{AgdaMultiCode}):
\begin{AgdaMultiCode}
\begin{code}[hide]
  postulate
    _ :
\end{code}
\begin{code}
       Set
\end{code}
\begin{code}[hide]
  postulate
    _ :
\end{code}
\begin{code}
       Set
\end{code}
\end{AgdaMultiCode}

\noindent Extra indentation (indentation, \texttt{AgdaMultiCode}):
\begin{AgdaMultiCode}
\begin{code}[hide]
  postulate
    Aaa :
\end{code}
\begin{code}
      Set
\end{code}
\begin{code}[hide]
  postulate
    Bbb :
\end{code}
\begin{code}
      Set
\end{code}
\end{AgdaMultiCode}

\end{document}
