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I am trying to learn how to use the $z$ notation to make complex numbers easier to work with. Instead of immediately changing $z$ to $x+yi$, is there a better strategy to make the problem less algebraically intense?

More specifically. I want to write the above express in the form $a+bi$. The end goal is to be able to parameterize the expression into $u(x,y), \text{ and } v(x,y)$ to get $u(x,y)+ v(x,y)i$

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    It would help if you gave more examples of what you are referring to. The example in the title is not particularly intense in either the '$z$' form or the '$x+iy$' form. – 2017-01-27
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    What do you mean by "make the problem less algebraically intense?" What is the problem you are trying to solve? – 2017-01-27
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    Sorry for my lack of clarity...I just added some more information... I want to parameterize the problem into u(x,y)+v(x,y)i – 2017-01-27
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    It's not too bad, just gotta divide by a complex number. – 2017-01-27

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You can simplify the fraction in complex numbers, then replace $z=x+yi\,$: $$ \frac{\bar{z}}{z+4} \cdot \frac{\bar z+4}{\bar z+4} = \frac{\bar z^2 + 4 \bar z}{\left|z + 4\right|^2} = \frac{\left(x-yi\right)^2 + 4 (x-yi)}{\left|x + 4 + yi\right|^2} = \frac{x^2-y^2+4x - (2xy+4y)i}{(x+4)^2 + y^2} $$

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    @Math4Life I just fixed a silly calculation error. It changed the end result, but not the underlying idea. – 2017-01-27