A quaternion Hadamard matrix is any $N \times N$ matrix $H$ which elements belong to the set $$\mathbb{H}=\{a+bi+cj+dk:a,b,c,d\in\mathbb{R}\}$$ of quaternions characterized by the equations: $i^2=j^2=k^2=ijk=-1$ that satisfies two conditions: $ |H_{mn}|=1$ for $m,n=1,2,...,N$ and $ HH^{\dagger}=N\mathbb{1}_N$ where $\mathbb{1}_N$ denotes the identity matrix of dimension $N$.
Some results in the subject of quaternion Hadamards and quaternion MUBs are included in the paper of Oleg Chterental and Dragomir Ž. Djoković [9].