The title says it all. This polynomial splits over $\mathbb{C}$ pretty obviously, $(t - \sqrt{i})(t + \sqrt{i})(t - i\sqrt{i})(t + i\sqrt{i}),$ but the matrix needs real entries, so I can't just make a diagonal matrix.
So my next idea is that $A^4 = -I,$ write $A^4 = B^2,$ where $B = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}.$ Then all I need to do is find $A$ such that $A^2 = B.$ Not possible (I think), probably because I started working with $2 \times 2$ matrices whereas the question does not give a size to he matrix, it just says "find a matrix with minimal polynomial ..."
This leaves me with questioning my approach, and that is why I've posted this question, how does one go about constructing a matrix given that it needs to satisfy a minimal polynomial, restricted to having elements in some field (preferably $\mathbb{R}$ or $\mathbb{C}$)? Solutions would be appriciated, but I am mostly interested in the procedure for constructing such matrices given a general minimal polynomial.
For fun, the subquestion associated with the original question is, "Show that the usual real linear map $v \to Av$ has no non-trivial subspace."