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I get:
sage: z = 1 + I
sage: sqrt(1/z)
sqrt(-1/2*I + 1/2)
If you want the real and imaginary parts, you may do:
sage: s = sqrt(1/z)
sage: s.real().simplify()
1/4*2^(3/4)*sqrt(sqrt(2) + 2)
sage: s.imag().simplify()
-1/4*2^(3/4)*sqrt(-sqrt(2) + 2)
