Identity Crisis

From high-school mathematics, we all know the complex cube roots of unity. Now here is a question:

How about cube roots does the identity matrix have?

Now, first of all let me define cube root of a square matrix. C is a cube root of A if C X C X C = A.

Of course, this seems like an obvious definition, but there are sophisticated definitions of what a cube root of a matrix is. For our purposes, let us stick with the simple definition.

And for simplicity, let us consider 2x2 matrices.

Clearly, the Identity Matrix is its own cube root.

But are there any others?

It depends on what all types of entries you allow in the matrix.

Consider the general case of a 2x2 matrix of complex numbers.

Now, I used Mathematica to find the following cube roots. Here is a cube root of (-1)  and c stands for any number.

Can you come up with some more?

(Hints: If A is a cube root of the identity what can you say about its transpose? What about its conjugate? Its inverse, if it exists?)