[adrian_b]

The form with quaternions from 1873 was not really correct, because at that time Maxwell did not understand yet the difference between vectors and pseudovectors (the "imaginary" components of a quaternion are the components of a pseudovector, not of a vector). This difference was clarified in the geometric algebra of Clifford, but both Maxwell and Clifford have died about at the same time and too early, so Maxwell did not have the opportunity to correct his treatise in a new edition.

I always wonder how physics could have evolved if both Maxwell and Clifford had not died in 1879. Many years later, Heaviside still did not understand the nature of vectors and pseudovectors (a.k.a. polar vectors and axial vectors, a.k.a. vectors and bivectors), so he has used extensively what he named as the "vector product", which is actually a pseudovector product, and not always in the right way.

On the other hand, the original version of the Maxwell equations was in integral form, not in differential form.

The integral form is applicable even when the equations of Heaviside do not exist and based on the integral form it is possible to derive forms of the equations that are valid even for bodies in movement, for which it is quite difficult to apply the Heaviside equations without obtaining erroneous results (because the Heaviside equations as normally presented in textbooks depend on additional unwritten assumptions, e.g. about the reference system, so their correct application e.g. to a motor is non-obvious).

The integral form given by Maxwell is the fundamental form of the equations, while the Heaviside differential form is a derived form whose applicability has serious restrictions and its only advantage is that it is easier to write and memorize by students.