Calling both basic and derived methods

In C #, no if the derived method is an override, but yes if it is marked as new. When using this design, you should be careful, as this is actually not what the consumer of your property would expect in most cases;

static class Program
{
    static void Main()
    {
        Base baseObject = new Derived();
        Derived derivedObject = new Derived();
        Console.Write(derivedObject.Test());
        Console.Write(baseObject.Test());
        Console.Write(((Base)derivedObject).Test());
    }
}

class Base
{
    public virtual int Test()
    {
        return 1;
    }
}

class Derived : Base
{
    public new int Test()
    {
        return 2;
    }
}