FWIW, I've been writing C# code for quite a time now, and the only case where I needed the new keyword was the following OO hierarchy scenario:

public abstract class Animal
{
    protected Animal(Animal mother, Animal father)
    {
        Mother = mother;
        Father = father;
    }

    public Animal Mother { get; }
    public Animal Father { get; }
}

public class Dog : Animal
{
    public Dog(Dog mother, Dog father)
        : base(mother, father)
    {
    }

    public new Dog Mother => (Dog)base.Mother;
    public new Dog Father => (Dog)base.Father;
}

Few remarks:

• No OO rule is broken, base classes still work fine.
• Properties are not marked virtual on purpose, they don't need to be.
• In fact we enforce more OO rules thanks to the new keyword: a dog cannot have any animal as a father/mother. It must be a dog.
• the new keyword is used because we cannot "override" a property (or method) just by changing its return type. new is the only way to have a nice OO hierarchy.