We denote the real quaternion algebra by $\mathbb{H}.$ In Zhang, Fuzhen. "Quaternions and matrices of quaternions." Linear algebra and its applications 251 (1997): 21-57., I'm having some trouble following.
Example 5.1. Let $$A=\begin{pmatrix}0&i\\j&0\end{pmatrix}.$$ Then the left eigenvalues of $A$ are $1$ and $i,$ whereas the right eigenvalues of $A$ are $1$ and all the quaternions in $[i]=\{gig^{-1}\mid g\in\mathbb{H},g\neq0\}.$
Theorem 5.4 (Brenner, 1951; Lee, 1949). Any $n\times n$ quaternion matrix $A$ has exactly $n$ (right) eigenvalues which are complex numbers with nonnegative imaginary parts.
Example 5.1 is probably the counterexample to Theorem 5.4 because $A$ has infinite right eigenvalues. I also see another example. For instance, the right eigenvalues of $$B=\begin{pmatrix}1-j+k&j+k\\1+i-j-k&i-j+k\end{pmatrix}$$ are $1$ and all the quaternions in $[i]$. I don't know if I misunderstood Theorem 5.4.
Also, I'm wondering why we do research first right eigenvalues and not left eigenvalues. I think left eigenvalues are closer to Linear algebra over fields. Thanks for all your support.