Can handmade pasta be more than just flour + eggs?
Traditional pasta such as you describe uses semolina flour and eggs, but many recipes also call for milk. Buttermilk or sour cream is often used in vareniki. Spaetzle recipes often call for nutmeg. Rice noodle recipes don't use either wheat flour or eggs, but use rice flour, water, and an additional starch such as tapioca. Some recipes call for butter or oils, some use different kinds of flour, some include leavening agents like baking powder. There are even recipes for sweet pastas. 

In short, there are many different ingredients used in pastas around the world, and you should feel free to experiment with your own choices. The key is to maintain the correct dough consistency. if you add a liquid ingredient, you may need a bit more starch, and vice versa.