In Morton L. Curtis's book Matrix Groups, it is stated that the orthonormality of the rows of a quaternionic matrix implies the orthonormality of the columns (Chapter 2 Proposition 4).
However, as the quaternion multiplication is non-commutative, I cannot see why this is true. All I could find is that the orthonormality of the rows of quaternionic matrix $A$ implies that the orthonormality of the columns of $\bar{A}$ which can be observed quite easily. Thus the question reduces to this:
If $x,y$ are orthonormal vectors with quaternionic coefficients, can we say that $\bar{x},\bar{y}$ are orthonormal?