Complex matrices column alignment on sign

If no further packages can be loaded you have to add the correct \medmuskip around the binary operators by hand

\documentclass{article}

\usepackage{amsmath}

\begin{document}

\[ % never use $$...$$ in LaTeX: https://tex.stackexchange.com/q/503/82917
\left[
\begin{array}{*{3}{r@{\mskip\medmuskip}c@{\mskip\medmuskip}l}}
64&+&2.828i & 1&+&1i      & 36&+&2.449i \\
9&+&1.732i  & 25&+&2.236i & 49&+&2.646i \\
16&+&2i     & 81&+&3i     & 4&+&1.414i 
\end{array}
\right]
\]

\end{document}

Why not simply three aligned environment inside a\bmatrix`?

Unrelated: for displayed equations, use the latex construct \[ ... \], note plain TeX $$ ... $$.

\documentclass[11pt, titlepage, oneside, a4paper]{article}
\usepackage{amsmath}

\begin{document}

   \[ \begin{bmatrix}
      \, \begin{aligned}
         64&+2.828i \\ 9&+1.732i \\ 16&+2i
       \end{aligned}
      & \begin{aligned}
       1&+1i \\ 25&+2.236i \\ 81&+3i
       \end{aligned}
       & \begin{aligned}
       36&+ 2.449i \\ 49&+ 2.646i \\ 4 & + 1.414i
       \end{aligned}\,
    \end{bmatrix} \]

\end{document} 

If you're willing to work with some other packages, you can capture the argument of every cell in a newly-defined column (say) I and then process (separating out the real/imaginary parts):

\documentclass{article}

\usepackage{amsmath,eqparbox,collcell,etoolbox}

\newcolumntype{I}{>{\collectcell\RpmI}c<{\endcollectcell}}
\makeatletter
\def\rplusi@#1+#2\relax{% R + Ii
  \eqmakebox[R][r]{$#1$} + \eqmakebox[I][l]{$#2$}%
}
\def\rminusi@#1-#2\relax{% R - Ii
  \eqmakebox[R][r]{$#1$} - \eqmakebox[I][l]{$#2$}%
}
\newcommand{\RpmI}[1]{%
  \def\abc{#1}% Store argument R +/- Ii
  \patchcmd{\abc}{+}{+}{%
    \expandafter\rplusi@#1\relax% R + Ii
  }{%
    \expandafter\rminusi@#1\relax% R - Ii
  }%
}
\makeatother

\begin{document}

\textbf{Works for real matrices:}
\[
  \left[\begin{array}{ccc}
    64 - 2.828i &  1 + 1i     & 36 - 2.449i \\
     9 + 1.732i & 25 - 2.236i & 49 + 2.646i \\
    16 - 2i     & 81 + 3i     &  4 - 1.414i
  \end{array}\right]
\]

\textbf{Workaround:}
\[
  \left[\begin{array}{III}
    64 - 2.828i &  1 + 1i     & 36 - 2.449i \\
     9 + 1.732i & 25 - 2.236i & 49 + 2.646i \\
    16 - 2i     & 81 + 3i     &  4 - 1.414i
  \end{array}\right]
\]

\end{document}