OpenSceneGraph uses header files without the ‘.h’ extension. Is there a way of getting Xcode to recognise the syntax as C++ and therefore allow me to have highlighting and jumping to definitions etc?
Eg: No highlighting:

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Yes. Toggle the right panel (the rightmost icon of the “View” section of the toolbar, arrow 1 in the screenshot) with your file selected. Select the document tab in the panel (arrow 2), then find “File Type” under “Identity and Type” (arrow 3). Choose “C++ header” in the dropdown list.
You will need to reopen the file for the syntax coloring to kick in.