Complex square matrices $A, B$ where $A$ and $B$ have a square root but $A+B$ doesn't

Question

Are there any matrices like this?

I think yes... choose $B$ to be all zeros except $1$ at an off diagonal component $(i,j)$ that affords $B$ a square root. Then choose $A$ to be a normal matrix where adding $1$ to the $(i,j)$ component makes it defective without a square root.

Of course this is not an explicit example, but I can't convince myself this situation couldn't happen.

Answer 1

Consider $$A=\begin{pmatrix} 1 & 1 \\ 0 & 0 \end{pmatrix} \quad \quad \text{and} \quad \quad B=\begin{pmatrix} -1 & 0 \\ 0 & 0\end{pmatrix}$$

You have $A^2=A$ and $(-iB)^2=B$, so both have a square root, but $$A+B=\begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$$

has no square root (because it is $2-$nilpotent).