> Maxwell was also not really taking advantage of them much and always split them up into scalar and vector part.

Did Maxwell actually use quaternions? If I recall correctly, at least in A Treatise on Electricity and Magnetism, quaternions were not actually used. Instead, he did most things in Cartesian coordinates, and all equations were applied to a vector's x, y, z components tediously. But many sources claimed Maxwell used quaternions, including quotes from Lord Kelvin. My reading on this part of history is limited, so my guess is that he did use them in personal research or in later papers. On the other hand, some other physicists of the same era used quaternions extensively, including applying them to Maxwell's electromagnetism, that is a sure fact...

Coincidentally, A Treatise on Electricity and Magnetism was written as an overview all electromagnetic phenomena as a whole, so it paid very little special attention to the generation and transmission of electromagnetic waves. Combining that with its difficult math, the book would puzzle physicists for another decade before they see the light from the book, and made it a rather curious period of history in electromagnetism.

> I've been trying to find these 4 equations in Heaviside's writing but so far have not been successful.

In 1885, Heaviside published Electromagnetic Induction and Its Propagation in The Electrician, and formulated what he called the "Duplex Form" of Maxwell's equation. This was a long series of papers published in several months, and later republished in Electric Papers, Volume I. Basically, following his physical intuition, he felt that electric and magnetic fields should be symmetric and generate each other, and that should be directly highlighted in equations.

The logic of the paper went like the following.

First, he started with a definition of electric current [1]:

C = kE
D = cE / 4π
Γ = C + D

in which, E denotes electric force, C denotes conduction current, k denotes specific conductivity constant, D denotes displacement current, and c denotes dielectric constant. Finally, Γ denotes true electric current, which is the sum of the conduction and displacement terms.

Next, a definition of magnetic current [2]:

B = µH
G = Ḃ / 4π = µḢ / 4π
G' = gH + µḢ / 4π

H denotes magnetic force, B denotes magnetic induction, µ denotes permeability, G denotes magnetic current, Ḃ and Ḣ are derivatives of B and H (Newton's notation). Hypothetically, suppose that magnetic monopoles exist (Heaviside did so), G' would denote the "true magnetic current", with an extra conduction term gH, where g is a constant similar to k.

Then, he introduced the concepts of divergence and curl, and their physical significance [3]. After more discussion and derivation, he finally wrote [4]:

curl (H - h) = 4πΓ = 4πkE + cĖ
-curl (e - E) = 4πG = 4πgH + µḢ

in which, e and H denote impressed electric and magnetic forces to take static fields into account. Finally, since magnetic monopoles don't exist, he made g = 0, but kept this term in the equations for symmetry and elegance. [0]

This is the core of Heaviside's Duplex Form of Maxwell's equations. one can clearly see the co-evolution of electric and magnetic fields, and is the precursor of the modern Maxwell's equations as we know today in its vector calculus formulation. As far as I know, his treatment of "physical" vectors as first-class objects is his original invention (independently invented by Gibbs as well), although the concepts themselves came from quaternions.

This is not a complete summary, as he continued his analysis in a series of publications.

A good book on this part of history is Oliver Heaviside: the life, work, and times of an electrical genius of the Victorian age, by Paul J. Nahin.

[0] So the claim "Maxwell's equations need modifications if magnetic monopole has been discovered" is historically inaccurate, it should rather be, "be restored to Heaviside's original form."

[1] Electric Papers, Volume I, Page 429, https://archive.org/details/electricalpapers01heavuoft/page/...

[2] Page 441: https://archive.org/details/electricalpapers01heavuoft/page/...

[3] Page 443: https://archive.org/details/electricalpapers01heavuoft/page/...

[4] Page 449: https://archive.org/details/electricalpapers01heavuoft/page/...