equation alignment in a two-column table

Use aligned and a nested aligned:

\documentclass{article}

\usepackage{amsmath}
\usepackage{newtxtext,newtxmath} % NOT times

\begin{document}

\begin{equation*}
\begin{aligned}
& \text{Addition}       && (a+bi)+(c+di)=(a+c)+(b+d)i \\
& \text{Subtraction}    && (a+bi)-(c+di)=(a-c)+(b-d)i \\
& \text{Multiplication} &&
    \begin{aligned}[t]
    z\cdot w &= r(\cos\theta+i\sin\theta)\cdot k(\cos\psi+i\sin\psi) \\
             &= rk\cos(\theta+\psi)+i\sin(\theta+\psi)
    \end{aligned} \\
& \text{Division}       &&
    \begin{aligned}[t]
    \frac{z}{w} &= \frac{r(\cos\theta+i\sin\theta)}{k(\cos\psi+i\sin\psi)} \\
                &= \frac {r}{k}[\cos(\theta-\psi)+i\sin(\theta-\psi)]
    \end{aligned}
\end{aligned}
\end{equation*}

\end{document}

Some notable points:

1. times is obsolete, better using newtxtext and newtxmath;
2. none of your \left and \right is useful, rather they're all harmful to spacing;
3. I changed the * into a middle dot;
4. I added the missing line for the multiplication;
5. Lowering “Multiplication” and “Division” to the middle of the two lines would make the text ambiguous (experiment the difference when you remove the two [t] positional arguments).

You can use a \phantom to align the second row for your Division title:

\documentclass{article}

\begin{document}

\noindent
\begin{tabular}{ @{} l l }
  Addition       & $(a + bi) + (c + di) = (a + c) + (b + d)i$ \\[\jot]
  Subtraction    & $(a + bi) - (c + di) = (a - c) + (b - d)i$ \\[\jot]
  Multiplication & $z \times w = r (\cos \theta + i \sin \theta ) \times k (\cos \psi + i \sin \psi )$ \\[\jot]
  Division       & $\frac{z}{w} = \frac{r (\cos \theta + i \sin \theta)}{k (\cos \psi + i \sin \psi)}$ \\
                 & $\phantom{\frac{z}{w}} = \frac{r}{k} [\cos (\theta - \psi) + i \sin (\theta - \psi)]$
\end{tabular}

\end{document}

with array and aligned:

\documentclass{article}
\usepackage{amsmath}
\usepackage{newtxtext,newtxmath} % NOT times

\begin{document}
\[\renewcommand\arraystretch{1.5}
\begin{array}{ll}
\text{Addition}       & (a+bi)+(c+di)=(a+c)+(b+d)i \\
\text{Subtraction}    & (a+bi)-(c+di)=(a-c)+(b-d)i \\
\text{Multiplication} & z\cdot w = r(\cos\theta+i\sin\theta)\cdot k(\cos\psi+i\sin\psi) \\
\text{Division}       & \begin{aligned}[t]
\frac{z}{w} &= \frac{r(\cos\theta+i\sin\theta)}{k(\cos\psi+i\sin\psi)} \\
            &= \frac {r}{k}\bigl[\cos(\theta-\psi)+i\sin(\theta-\psi)\bigr]
                        \end{aligned}
\end{array}
\]
\end{document}