Solving a complex equation involving conjugates with Mathematica

[GargleBlast42]

How can one obtain (using Mathematica) all the solutions of an equation involving conjugates like for example $$z \bar{z}=1$$. Although this equation is very simple, it seems that Mathematica is not able to handle it. I obtain:

Solve[z Conjugate[z] == 1, z]
InverseFunction::ifun: Inverse functions are being used. Values may be lost for multivalued inverses. >>
Solve::tdep: The equations appear to involve the variables to be solved for in an essentially non-algebraic way. >>
Solve[z z^\[Conjugate]\[LongEqual]1,z]

Of course I'm not interested in this particularly simple equation, but of an equation involving many terms, which can't be, unfortunately, done by hand in a reasonable amount of time.
Any ideas?

 

[Dale]

When Solve fails, especially with the non-algebraic error, I usually use Reduce.