How/why are the $mathcal Z$-transform and unit delays related?

The $\mathcal Z$-transform uses the same notation as the unit delay $z^{-1}$, but in $\mathcal Z$-transform $z$ is a complex number.

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  What's the relation between the $\mathcal Z$-transform and the unit delay concept?

  
  
  


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  Also, is there a complex number interpretation of unit delay?