How to apply other than trivial dimensions?

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I only used natural dimensions so far but I understand there could be

  • Negative dimension with application e.g. dimension -2 definition and usage
  • Non-integer dimension with application e.g. dimension ½ defintion and usage
  • Imaginary dimension with application e.g. dimension 2i definition and usage

Could you please examplify why such dimension(s) like above should exist, when mathematically I suppose one could argue than one or more of the above mentioned are a consequence from definitions of what a dimension is. In complex analys something called the Riemann sphere is discussed which introduces a 4th axis, is that axis also "trivial" ?