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Do you mean the yellow liquid crystal segments that blink into life to form the number "24"? Or the multiple live frames on the screen at once? The latter should not be too difficult to replicate using PC video editing software - it would probably have to be a fairly powerful PC to do the edit, but once assembled should play fine.
The 24 LCD segment build is actually quite subtle, the colouring of the segments is not entirely even giving it an authentic slightly "burned in" LCD effect. I suspect this would be the biggest challenge - as each segment is a geometric shape it would be relatively easy to animate, rendering it in a way that it wouldn't look like a cartoon would be more challenging.
You can probably achieve it in Blender, which is an open-source animation suite, give youself some time to learn though, the concepts and software are not trivial. http://www.blender.org/cms/Home.2.0.html