Real and complex tetration

[Daniel]

My approach to fractional tetration involves the complex numbers. Others have found fractional tetration constrained to the real numbers to be more natural. Which tetration methods require complex numbers and which only need real numbers? For example I consider Kneser as real based because fractional tetration of f(x) where \( f(f(x))=e^x \) is real even though the proof uses complex numbers.