A universe with fully symmetric electromagnetism? Although several extensions to the Standard Model predict the possible existence of magnetic monopoles, their expected properties are rather significantly different from those of the electrically-charged particles we all know and love--the coupling constant for monopoles is huge, if they exist they are experimentally constrained to huge masses, and as such would not form interesting complex structures like normal baryonic matter held together by electric charges.
It would seem to me more "elegant" if electromagnetism were in fact fully symmetric, such that charged leptons & quarks all come in electric / magnetic pairs, and coupling constants and masses were more similar such that, although the results need not be absolutely identical, both versions of matter can form something like atoms and molecules, etc.
So, in the spirit of thought experiments like A Universe Without Weak Interactions, or Greg Egan's alternate-metric universes (Dichronauts & Orthogonal), is it possible to construct a fully symmetric theory with co-existent electric and magnetic matter? How off-the-rails do things get if we just set $\alpha$ to 1 (so it's reciprocal is also 1)? If it is possible, what are the minimal other changes necessary to preserve a universe with recognizable chemistry? And given the existence of parallel electric and magnetic quarks, would electric and magnetic baryons still exhibit strong force interactions with each other, resulting in dyonic nucleii, or do we end up with two completely isolated parallel periodic tables?
 A: The attempt to have electric and magnetic charges in the same theory, led to GUT monopoles, the lattice of dyon charges, SL(2,Z) duality, the Seiberg-Witten model of confinement, and S-duality in string theory. You could even say it has converged on pure math via its connections with the Langlands program (e.g. the Langlands dual group). It's a major theme in modern theoretical physics, discussed e.g. here. 
Trying to identify a field theory which is like the standard model, but with magnetically charged almost-copies of all the standard model fermions, that are as similar in mass and couplings as possible... I leave this task to others, for whom it may have more intrinsic appeal. But I will mention three things:
1) The work of Robert Finkelstein on "the SLq(2) extension of the standard model" (latest paper). 
2) The work of Tanmay Vachaspati on mimicking the standard model with monopoles (first paper). 
3) The most fruitful way to tackle the questions posed, may be in the context of Vafa's "swampland" research program, which aims to identify what kinds of field theory cannot be found in the string theory "landscape". There are classic results constraining the possibilities of magnetic charge, but there are probably new constraints to be discovered, if one assumes string theory as a framework. 
