What would the world look like if muons were stable? Since they are a sort of 'heavy electrons' also 'muonic atoms' could form. These have ben also produced in the laboratory. However, they are unstable. But I was wondering what consequences it would have for the world we live in if, besides the 'normal atoms' also muonic atoms would exists? Or, also, what if normal atoms would have been replaced by muonic atoms? Could chemical bonds form at all?
 A: Using this venue since there is no space for a long comment.
I assume you want stable muons on top of stable electrons, right? In that case the ratio
$$
m_\mu/m_e= 207 , \qquad  ~  m_p/m_\mu =8.9
$$
informs all else. The typical "Bohr radius"  for a muon, $a_\mu$,  is then 1/207 times smaller than our Bohr radius, $a_0$,
$$
a_\mu= a_0 / 207,
$$
so you'd have interesting two-tier hybrid atomic and molecular physics, with intriguing new possibilities...
You'd  have a nucleus, a muonic shell nearby, and an electron shell further out...  But the muonic shell might appear to merge into a fatter nucleus, resulting in a fat nucleus of different charge,
an exotic atom,

Hydrogen-4.1, also known as "neutral muonic helium",cf this is similar to helium because it has 2 protons and 2 neutrons, but one of its electrons is replaced by a muon. Since the orbital of the muon is very near the atomic nucleus due to its larger mass, that muon can be considered as a part of the atomic nucleus. The atomic nucleus is composed of 1 muon, 2 protons and 2 neutrons, and only one electron orbiting, so it can be considered as an exotic isotope of hydrogen. A muon's weight is 0.1u, hence the name Hydrogen-4.1(4.1H). The Hydrogen-4.1 atom can react with other atoms. It behaves like a hydrogen atom rather than a helium atom.cf this

You'd certainly need a new notation from the above: Not only the atoms would be different/richer, but there would be muon-bound non-neutral molecules within electron-atoms...
I gather the two spectroscopies would be easy to separate, but maybe you'd need an atomic expert to address that. Spin-spin interactions would be a new intriguing ambit of mischief...
A: One dramatic consequence is that the hydrogen molecule H2 would not be stable.  (See "muon catalyzed fusion").  Because of the vastly reduced size of atoms, there would be a vastly reduced barrier to nuclear fusion.  Stars would burn out very quickly.  Probably life would be impossible.
A: In such a world the periodic table would probably have to have another axis on it (3D). Two atoms with identical nucleuses but one with a largely muon cloud surrounding it and the other with a largely electron cloud would be very different. Roughly speaking one might expect that the Muons would stay close to the nucleus and not get involved much in chemistry.
If the number of Muons and electrons were equal overall (so half of all electons are tuned into stable muons) then I would expect most natural atoms in this world have electrons and muons in equal numbers, that would allow the atom to be electrically neutral while using only half as many orbitals as the atoms we are used to. So it would use the first half, the lower energy ones. (You can't put two electons of the same spin in the same orbital, but you can put two electrons of opposite spins in an orbital and then still put two muons of opposite spin into a similar orbital. (The muon version of the same orbital)).
