Quaternions form a set of four global but not unique parameters, which canrepresent three-dimensional rotations in a non-singular way. They arefrequently used in computer graphics, drone and aerospace vehiclecontrol. Floating-point quaternion operations (addition, multiplication,reciprocal, norm) are often implemented "by the book". Although allusual implementations are algebraically equivalent, their numericalbehavior can be quite different. For instance, the arithmeticoperations on quaternions as well as conversion algorithms to/from rotation matricesare subject to spurious under/overflow (an intermediate calculationunderflows or overflows, making the computed final result irrelevant,although the exact result is in the domain of the representablenumbers). The goal of this paper is to analyze and then proposeworkarounds and better accuracy alternatives for such algorithms.