I agree the imaginary parts should be zero. I do not know why they are not zero. But this is what I found, too small to put in comment.
First, Matlab does give zero for the exact same input:
format long g
mat = [2, 2.161209223472559 + 1.682941969615793*1j;
2.161209223472559 - 1.682941969615793*1j, 2]
inv(mat)
-0.570919803469126 + 0i 0.616938572560308 + 0.480412449271497i
0.616938572560308 - 0.480412449271497i -0.570919803469126+0i
You can get same output in Mathematica by doing the direct computation itself without calling Inverse
foo[x11_, x12_, x21_, x22_] :=
Module[{}, {{x22, -x12}, {-x21, x11}}/(x11*x22 - x21*x12)]
foo[2, 2.161209223472559 + 1.682941969615793 I,
2.161209223472559 - 1.682941969615793 I, 2]
It is an exact zero for the complex part:
And can see it match the Matlab output. Even compiled version did not resolve the issue (even though told it to run in hardware floating point)
cf = Compile[{{x11, _Complex},{x12, _Complex},{x21, _Complex},
{x22,_Complex}},
Inverse[{{x22, -x12}, {-x21, x11}}], RuntimeOptions -> "Speed"
];
cf[2, 2.161209223472559^1 + 1.682941969615793^1 I,
2.161209223472559^1 - 1.682941969615793^1 I, 2]
If we do break the inverse to 2 parts, and do one by 'hand' and then use Det only, then now the accuracy improves a little, and now it is of order 10^-17
foo1[x11_, x12_, x21_, x22_] :=
Module[{}, {{x22, -x12}, {-x21, x11}}/Det[{{x11, x12}, {x21, x22}}]]
foo1[2, 2.161209223472559 + 1.682941969615793 I,
2.161209223472559 - 1.682941969615793 I, 2]
Here is Maple 2015 result also:
mat:=<<2|2.161209223472559 + 1.682941969615793*I>,
<2.161209223472559 - 1.682941969615793*I|2>>;
LinearAlgebra[MatrixInverse](mat);
