I'm not sure how to covert those sqrt expressions into what you want.
However, you can see that they are what you're expecting:

sage: A = matrix([[-1,-1+I],[1,0]])
sage: evs = A.eigenvalues(); evs
[(-sqrt(4*I - 3) - 1)/2, (sqrt(4*I - 3) - 1)/2]
sage: map(CC, evs)
[-1.00000000000000 - 1.00000000000000*I, 1.00000000000000*I]

--Mike

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