How do you center multiple equations that have multiple steps?

Here are three different suggestions using either array or alignedat:

\documentclass{article}
\usepackage{geometry}
\usepackage{amsmath}

\begin{document}

\[ 
\begin{array}{llll}
N(1)=N(1+0i) & N(-1)=N(-1+0i)   & N(i)=N(0+1i) & N(-i)=N(0-1i)   \\
N(1)=1^2+0   & N(-1)=(-1)^2+0^2 & N(i)=0^2+1^2 & N(-i)=0^2+(-1)^2 \\
N(1)=1       & N(-1)=1          & N(i)=1       & N(-i)=1
\end{array}
\]

\[ 
\begin{array}{cccc}
N(1)=N(1+0i) & N(-1)=N(-1+0i)   & N(i)=N(0+1i) & N(-i)=N(0-1i)   \\
N(1)=1^2+0   & N(-1)=(-1)^2+0^2 & N(i)=0^2+1^2 & N(-i)=0^2+(-1)^2 \\
N(1)=1       & N(-1)=1          & N(i)=1       & N(-i)=1
\end{array}
\]


\begin{alignat*}{4}
N(1)&=N(1+0i) &\quad N(-1)&=N(-1+0i)   &\quad    N(i)&=N(0+1i)  &\quad N(-i)&=N(0-1i)   \\
N(1)&=1^2+0   &      N(-1)&=(-1)^2+0^2 &         N(i)&=0^2+1^2  &      N(-i)&=0^2+(-1)^2 \\
N(1)&=1       &      N(-1)&=1          &         N(i)&=1        &      N(-i)&=1
\end{alignat*}

\end{document}

I propose this layout:

\documentclass{article}
\usepackage[utf8]{inputenc}%
\usepackage{geometry} 
\usepackage[table, svgnames]{xcolor}
\usepackage{mathtools}
\colorlet{shadecolor}{Gainsboro!50! Lavender}
\newcommand*\shadebox[1]{%
\colorbox{shadecolor}{\hspace{1em}$\displaystyle #1 $\hspace{1em}}}
\begin{document}


\begin{equation}
\begin{gathered}
\begin{aligned}
 N(1) & =N(1+0i)\\
  & =1^2+0^2
\end{aligned} \\[1.5ex]
\shadebox{N(1) = 1}
\end{gathered}
  \qquad
\begin{gathered}
\begin{aligned}
 N(-1) & =N(-1+0i) \\
& =(-1)^2+0^2
\end{aligned} \\[1ex]
 \shadebox{N(-1) = 1}
\end{gathered}
 \qquad
\begin{gathered}
\begin{aligned}
 N(i) & =N(0+1i) \\
  & =0^2+1^2
\end{aligned} \\[1ex]
\shadebox{ N(i) = 1}
\end{gathered}
 \qquad
\begin{gathered}
\begin{aligned}
 N(-i) & =N(0-1i) \\
  & =0^2+(-1)^2
\end{aligned} \\[1ex]
 \shadebox{N(-i) = 1}
\end{gathered}
\end{equation}

\end{document} 

For example I would have used the tables in sequence (but it's just a personal taste) to give a touch of vitality. In this case I have used the booktabs package to have \toprule and \bottomrule.

\documentclass[a4paper,12pt]{article}
\usepackage{array}
\usepackage{booktabs}

\begin{document}
\[
\setlength\arraycolsep{0pt}
\renewcommand\arraystretch{1.25}
\begin{array}{r @{{}={}} l}
\toprule
N(1) &N(1+0i)   \\
N(-1)&N(-1+0i)  \\
N(i) &N(0+1i)   \\
N(-i)&N(1+0i)   \\  
\bottomrule
\end{array}
\quad\Rightarrow\quad
\begin{array}{r @{{}={}} l}
\toprule
N(1) &1^2+0     \\
N(-1)&(-1)^2+0^2\\
N(i) &0^2+1^2   \\
N(-i)&0^2+(-1)^2\\  
\bottomrule
\end{array}
\quad\Rightarrow\quad
\begin{array}{r @{{}={}} l}
\toprule
N(1) &1 \\
N(-1)&1 \\
N(i) &1 \\
N(-i)&1 \\  
\bottomrule
\end{array}
\]
\end{document}