strange math

I am NOT a mathematician.
What I am about to write may be well known or be TOTAL NONSENSE.
So please bear with me.

There are two separate but otherwise identical complex systems.
I: a+bi
J: a+bj

Now combine the two systems and get

I+J:a+bi+cj

There may be many rules to resolve numeric combinations. Here is one example

ai+bj  =(a+b)i if a is greater than b
          (a+b)j if b is greater than a  ... otherwise 0

aixbj=axb

Any comments?

Oh come on, I get tired of receiving a reply giving ample references to studies of these equations, or to tell me that by my rules the whole thing reduces to nothing and is nonsense. So I will drop the other shoe anyway and describe the idea that strange math would be of interest to science fiction writers. Consider the plot of the following science fiction story:

A physicist discovers strange math and realizes that it is the key to accessing alternate universes. He/she postulates that I (I:a+bi) is valid in our normal universe. However J (J:a+bj) is a characteristic of the alternate universe. So he/she builds a structure that supports I+J with small and equal constants. When a person from normal universe enters the structure and increases the constant b, on exit from the structure he is in alternate universe. You get the idea.

Hi Jizba,

yes I get your idea as you summarized it in the second post...what I'm curious about, is how do you deal with the inherent differences in the living being of one universe vs another wrt long-term survivability?

Thanks for the response. As far as the science fiction analogy, I suppose it would be
the writer's option to resolve it. For example the alternate universe may have so different physics that no stars have ever formed there. But seriously, my analogy was intended to highlight the idea that a simple number can be made to have totally unexpected consequences. For example a blood test on a patient that changed from a normal reading to an anomalous value will cause the doctor to prescribe a totally different treatment.