I would like to be able to display the caret at all times in the SyntaxEditor, even when the SyntaxEditor focus is lost. However, I haven't found a property in the documentation about keeping the caret visible. I've seen some examples with setting FocusManager.IsFocusScope="True" on a a standard WPF TextBox but I don't think it can be applied to the SyntaxEditor. I tried it and the caret still disappear after losing focus.
If I want to achieve this, does my only solution is to create, in the same fashion as your AdornmentsIntraTextNotes example, an intra-space adorner that will show a caret look-alike (doesn't need to blink) and hook on SyntaxEditor LostFocus/GotFocus to make the adorner show/hide?