Marcus' function would be well defined if we agreed to use trailing nines wherever the decimal expansion is terminating, this should of course have been specified.
How could that possibly be a bijection? Obviously, z_1=a+ib is mapped to the same point as z_2=a z_1, so it is not an injection.
Marcus has already provided a valid bijection, his "decimal merging" is the classical example of this. Notice how it is also valid in \mathbb{R}^n.
I.e. the multiplicity is the power of the term with the largest negative power in the laurent series of the function?
Does this also mean that an isolated/(essential?) singularity is a pole with infinite multiplicity?