As the title states, I would like to declare an indexer object this[int index] in an abstract class as an abstract member.
Is this possible in any way? Is it possible to declare this in an interface too?
Of course:
public abstract class ClassWithAbstractIndexer 
{
    public abstract int this[int index]
    {
        get;
        set;
    }
}
A simple example:
public interface ITest
{
    int this[int index] { get;  }
}
public class Test : ITest
{
    public int this[int index]
    {
        get { ... }
        private set { .... }
    }
}
Several combinations of private/protected/abstract are possible for get and set
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