Solve a 2x2 Eigensystem
Python 2, 198 bytes
a,b,c,d=input()
H=(a+d)/2
D=(H*H-a*d+b*c)**.5
X,Y=H+D,H-D
p,q,r,s=[[1,0,0,1],[b,X-a,b,Y-a],[X-d,c,Y-d,c]][2*(c!=0)or(b!=0)]
A=abs
V=A(A(p)+A(q)*1j)
W=A(A(r)+A(s)*1j)
print[X,Y],[[p/V,q/V],[r/W,s/W]]

Input is a flat list of 4 complex numbers via STDIN, e.g.
[0.0+0j, 0.4+0j, -0.1+0j, -0.4+0j]

Note that Python uses j instead of i for complex numbers.
Output is two lists, the first containing the eigenvalues and the second containing the eigenvectors, e.g.
[(-0.2+0j), (-0.2+0j)]
[[(0.8944271909999159+0j), (-0.4472135954999579+0j)], [(0.8944271909999159+0j), (-0.4472135954999579+0j)]]

(newline inserted for clarity)