About type extensions. What you mean is subclassing in object-oriented languages. No, you cannot do that in Julia, but nor can you do that in almost any other functional language. Most modern languages eschew subclassing because it tends to bring more bad than good.

The approach in Julia and other non-OOP languages is typically to use composition rather than inheritance. If you struggle with organizing your code in Julia because you have a very OOP mindset about how to write code, I would recommend the book "Design Patterns and Best Practices with Julia" by Tom Kwong. I also cover this in my own book "Julia for Beginners" https://leanpub.com/julia-for-beginners

I do various programming examples in an OOP like style and functional style.

I don't know what you want to achieve with subclassing. Depending on the particular problem there might be much easier solutions in Julia. But the most straightforward mimicking of subclassing would be to use composition using the @forward macro from the Lazy.jl package.

You embed one type within another and then you list methods you want forwarded to the embedded type. Essentially the methods you are not overriding.