Centering the align environment on the middle column

Removing the first alignment character and adding a few \phantoms should do the trick. Alternatively you can use alignat:

\begin{document}
\begin{align}
    (x-1)^2 &= -1    \\
        x-1 &= \pm i \\
          x &= 1 \pm i
\end{align}

\begin{align}
    \phantom{\implies}    (x-1)^2 &= -1    \\
    \implies\phantom{()^2}    x-1 &= \pm i \\
    \implies\phantom{({}-1)^2}  x &= 1 \pm i
\end{align}

 \begin{alignat}{3}
             && (x-1)^2 &= -1    \\
    \implies &&     x-1 &= \pm i \\
    \implies &&       x &= 1 \pm i
 \end{alignat}
 \end{document}

If you want exactly the same alignment as before, but this time with \implies, this does the trick:

\documentclass{article}
\usepackage{amsmath}
\begin{document}
    \begin{align}
    (x-1)^2 &= -1     \\ x-1 &= \pm i  \\ x &= 1 \pm i
    \end{align}

    \begin{alignat}{3}
    && (x-1)^2 &= -1      && \phantom{\implies} \\ \implies &&     x-1 &= \pm i   && \\ \implies &&       x &= 1 \pm i &&
    \end{alignat}
\end{document}

The idea here is to duplicate the width of \implies on the left with a \phantom{\implies} on the right, thereby evenly spacing the equation/environment again.