What are the restrictions on Solve for complex variables?

Questions
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1

I don’t understand why Reduce works for this problem but Solve doesn’t.

In[15]:= Solve[{Abs[z] == 1, Abs[z - 1/2] == 1}, z, Complexes]

Out[15]= {}

In[17]:= Reduce[{Abs[z] == 1, Abs[z - 1/2] == 1}, z]

Out[17]= z == 1/4 - (I Sqrt[15])/4 || z == 1/4 + (I Sqrt[15])/4
2

Solve gives the solution if you add MaxExtraConditions → All

Solve[{Abs[z] == 1, Abs[z - 1/2] == 1}, z, Complexes, 
 MaxExtraConditions -> All]

Solve does not specify conditions in the solution, so quite puzzling for me

3

Thanks. The documentation says

Solve gives generic solutions only. Solutions that are valid only when continuous parameters satisfy equations are removed. Additional solutions can be obtained by using nondefault settings for MaxExtraConditions.

which explains it.