QR algorithm for "general" square matrices 
*

*Can QR algorithm find repeat eigenvalues (https://en.wikipedia.org/wiki/QR_algorithm) ? 
I.e. does it support the case when not all N eigenvalues for real matrix N x N are distinct?

*How to extend QR algorithm to support finding complex eigenvalues ?

*Is it possible to extend QR algorithm to work with not full rank matrices ?

Summary
The QR Decomposition resolves all of your matrix types.

a) Repeated eigenvalues
The three eigenvalues of $\mathbf{A}$ are
$$
 \lambda \left( \mathbf{A} \right) = \left\{ 1, 1, 1 \right\}
$$
The decomposition is 
$$
\begin{align}
\mathbf{A} &= \mathbf{Q\,R} \\
% A
\left[
\begin{array}{ccc}
 1 & 1 & 1 \\
 0 & 1 & 1 \\
 0 & 0 & 1 \\
\end{array}
\right]
&=
% Q
\left[
\begin{array}{ccc}
 1 & 0 & 0 \\
 0 & 1 & 0 \\
 0 & 0 & 1 \\
\end{array}
\right]
% R
\left[
\begin{array}{ccc}
 1 & 1 & 1 \\
 0 & 1 & 1 \\
 0 & 0 & 1 \\
\end{array}
\right]
%
\end{align}
$$

b) Complex eigenvalues
The three eigenvalues of $\mathbf{A}$ are
$$
 \lambda \left( \mathbf{A} \right) = \left\{ 1 + 2 i, 1 - 2 i \right\}
$$
The decomposition is 
$$
\begin{align}
\mathbf{A} &= \mathbf{Q\,R} \\
% A
\left[
\begin{array}{cr}
 3 & -2 \\
 4 & -1 \\
\end{array}
\right]
&=
% Q
\frac{1}{5}
\left[
\begin{array}{rc}
 3 & 4 \\
 -4 & 3 \\
\end{array}
\right]
% R
\left[
\begin{array}{cr}
 5 & -2 \\
 0 & 1 \\
\end{array}
\right]
%
\end{align}
$$

c) Matrices with rank defect
The following matrix has rank $\rho = 1$, therefore the matrix has both a row and a column rank defect.
The decomposition is 
$$
\begin{align}
\mathbf{A} &= \mathbf{Q\,R} \\
% A
\left[
\begin{array}{rrr}
 1 & -1 & 1 \\
 -1 & 1 & -1 \\
\end{array}
\right]
&=
% Q
\frac{1}{\sqrt{2}}
\left[
\begin{array}{r}
 1 \\ -1 \\
\end{array}
\right]
% R
\left[
\begin{array}{crc}
 \sqrt{2} & -\sqrt{2} & \sqrt{2} \\
\end{array}
\right]
%
\end{align}
$$
