I am not a lawyer, so take the following with an appropriate dosage of salt:
As I understand it, you cannot copyright a user interface, as determined in Apple v. Microsoft, primarily because it has a functional aspect, specifically: enabling users to interact with a computer. You can, however, file a design patent on your GUI.
According to Wikipedia, copyright only applies if the protected design has no functional/practical utility, which is where design patents step in:
The copyrighted artistic expression must either have no substantial
practical utility (e.g. a statue) or be separable from the useful
substrate (e.g. picture on a coffee mug).
Design patents, on the other hand, cover the ornamental aspects of
functional items from being infringed. One does not have to show that
the infringing item was copied from the original. Thus a design that
was arrived at independently can still infringe a design patent.
Note, however, that design patents are extremely narrow, and a design patent will cover only the exact or almost exact appearance of that GUI, down to the smallest graphical minutiae of the GUI elements. One can potentially avoid infringing a design patent simply by changing dimensions, text content and/or relative placements of GUI elements. Infringing design patents of any practical intricacy through independent invention is rather improbable.
As an aside, Apple learnt an important lesson from its loss against Microsoft, the repercussions of which can be seen in its IP strategy today. For instance, instead of relying only on design patents, it also protected various functional aspects of the UI with utility patents, such as the over-scroll bounceback and touch input heuristics. These cover the technical aspects that enable the functioning and behavior of the GUI rather than its appearance. Hence, if your GUI includes, for instance, any custom elements whose functionality or behavior is novel and non-obvious, that may be eligible for utility patent protection.