nullable object must have a value

You should change the line this.MyDateTime = myNewDT.MyDateTime.Value; to just this.MyDateTime = myNewDT.MyDateTime;

The exception you were receiving was thrown in the .Value property of the Nullable DateTime, as it is required to return a DateTime (since that's what the contract for .Value states), but it can't do so because there's no DateTime to return, so it throws an exception.

In general, it is a bad idea to blindly call .Value on a nullable type, unless you have some prior knowledge that that variable MUST contain a value (i.e. through a .HasValue check).

EDIT

Here's the code for DateTimeExtended that does not throw an exception:

class DateTimeExtended
{
    public DateTime? MyDateTime;
    public int? otherdata;

    public DateTimeExtended() { }

    public DateTimeExtended(DateTimeExtended other)
    {
        this.MyDateTime = other.MyDateTime;
        this.otherdata = other.otherdata;
    }
}

I tested it like this:

DateTimeExtended dt1 = new DateTimeExtended();
DateTimeExtended dt2 = new DateTimeExtended(dt1);

Adding the .Value on other.MyDateTime causes an exception. Removing it gets rid of the exception. I think you're looking in the wrong place.