I'll second that ejeffrey's statement is correct.
For a conservative field like the electric field in electrostatics you can define a potential whose gradient is the field. You can add a constant to the potential without changing its gradient.
However, if you have changing magnetic fields you can not define electric potential because the electric field is not conservative in that case. A closed integration path enclosing a changing magnetic field will measure a voltage, even though you start and end in the same point!
The constant has to correspond to a real property of the universe. It is not a number that anyone can arbitrarily make up. For the single object isolated in space, it would mean that the universe would have to have a non-zero charge. Not being a physicist, I cannot be sure that there is not a significant non-zero charge, but if there were it would have a lot of consequences.
If the constant is non-zero, it would be the equivalent to enclosing the universe in a conductive sphere charged to a voltage. If the universe does behave this way, it would mean that an uncharged hydrogen molecule will generate a local electric field that should be measurable.
The universe has been expanding since the big bang, so this universe charge will be continually decreasing towards zero. If there is a signifigant universe charge/voltage now, then it means that early in the big bang, the voltage of the universe would have been extreme, and this would have been a major factor in the way the universe has expanded.
My guess is that this universe voltage constant is probably so low it is insignifigant, but I admit I do not know for sure. I suspect that if you look at the known size of the universe and calculate a voltage based in a possible charge unbalance, the resulting voltage will still be below anything we could detect. So I still stand by my statement that an isolated charged body in space has a measurable absolute voltage.
Whatever the voltage of the universe is, it is nonsensical to suggest that the field around a charged isolated object in space is neither measurable or constant. It does not depend on any arbitrary constant. The field is a real physically property of space around the charged object, not a mathematical abstraction that can be altered with imaginary constants.
Or in practical terms: if you set your multimeter to AC volts, put one lead through the winding area of a transformer and short the tips together you will see a voltage!
You don't use a multimeter to measure the voltage of a charged object, as a multimeter draws current. You would need an electrometer, or any other method to examine the electric field such as the behavior of a charged particle as it passes though the field.
Richard.