is $1^z=1$ for all complex values of $z$? i would like to see if   $1^z=1$   is valid for all complex variable  $z$,first of all  you can rewrite above  equation as
$1^{a+b*i}=e^0$
here i think that instead of $+$ sign, we may take  take complex conjugate  form or $-$ sign.from above equation  we can  get
$1^a *1^{b*i}=e^0$
or  $1^{b*i}=1$
here i am assuming that  $a$  is real,otherwise  if we  have complex variable in power  complex variable,result will be undefined,so now question is:  could you conclude that
$1^{b*i}=1$?
generally  i  know that $i^0=1$
so
$i^{0*b*i}=1$
we get identity ,but is this right?
i would like to show you following article
http://www.cut-the-knot.org/do_you_know/complex.shtml
with title :
Complex number to a complex power may be real
what could i say above equation?
also  we  may introduce  some useful well known  identities,but which one could be  relevant for this  case?thanks very much

It depends on the particular branch of the complex logarithm you choose, for example:
$$1^i=e^{i\operatorname{Log}1}=e^{i\log|1|-\arg 1}=e^{-\arg 1}\in\{e^0=1\;,\;e^{-2\pi}\;,\;e^{2\pi}\;,\;e^{-2\pi}\,,\ldots\}$$
