The case serial number is of no use for determining the value of an Elgin pocket watch.
In the picture below you will see an Elgin watch and the serial number is at the top of this watch and on the actual movement.
Chris' explanation isn't as detailed as Kent's, but it includes pictures which can be very helpful.
The serial numbers were allocated in "runs" or blocks of identical grade watches.
Starting in the late 1930s, Elgin added a letter prefix to the serial numbers of some watches and around this time, Elgin also started to mark the grade number on some of the movements in addition to the serial number.
The Online Database for the Elgin Watch Company can tell you a great deal of information about your watch, based on just the serial number off the movement.
(The "movement" is also sometimes called the "watch works".) Important!
While their altruism was vital to the war effort, Elgin’s patriotism ironically opened an opportunity for the Swiss.
By 1964, after a Mid-Century decade that saw the rise of the elite “Lord and Lady Elgin” series, the original Elgin factory closed.
Never consider buying an antique Elgin pocket watch without first obtaining the pocket watch serial numbers.
There are usually two numbers and one can be found on the case and the most important one is located on the actual movement.
There would be no logical other reason to try and remove a serial number.