Charcoal, 25 bytes

⭆¹ΣＥ∕Ｘ⊘⊕×⟦¹±¹⟧₂⁵Ｎ₂⁵⎇μ⁻⁰λλ 

Try it online! Link is to verbose version of code. Explanation:

 ⟦¹±¹⟧ Literal list `1, -1`  × Vectorised multiply by ₂ Square root of ⁵ Literal `5` ⊕ Vectorised increment ⊘ Vectorised halved Ｘ Vectorised raised to power   Ｎ Input as a number ∕ Vectorised divide by   ₂ Square root of ⁵ Literal `5` Ｅ  ⎇μ⁻⁰λλ Negate the second element Σ Sum the elements ⭆¹ Stringify 

Charcoal doesn't really support complex numbers, but as it happens the Power of a negative number to a floating-point value will return a complex result; I just have to be careful not to use operations such as Negate and Cast where Charcoal will get confused as to which overload to use.

Charcoal, 25 24 bytes

⭆¹ΣＥ∕Ｘ⊘⊕⟦₂⁵±₂⁵⟧Ｎ₂⁵⎇μ⁻⁰λλ 

Try it online! Link is to verbose version of code. Explanation:

 ⁵  Literal `5`  ₂ Square root  ⁵  Literal `5`  ₂  Square root    ± Negated ⟦ ⟧ Make into list ⊕ Vectorised increment ⊘ Vectorised halved Ｘ Vectorised raised to power Ｎ  Input as a number ∕ Vectorised divide by ⁵ Literal `5` ₂ Square root Ｅ ⎇μ⁻⁰λλ  Negate the second element Σ Sum the elements ⭆¹ Stringify 

The version of Charcoal on TIO doesn't really support complex numbers, but as it happens the Power of a negative number to a floating-point value will return a complex result; I just have to be careful not to use operations such as Negate and Cast where Charcoal will get confused as to which overload to use.

18 bytes using the newer version of Charcoal on ATO:

Ｉ↨∕Ｘ⊘⊕⟦±₂⁵₂⁵⟧Ｎ₂⁵±¹ 

Attempt This Online! Link is to verbose version of code. Explanation: The version of Charcoal on ATO can use "base -1" conversion to take the difference of a list of two complex numbers saving 5 bytes and also can directly cast a complex number to string saving a further 1 byte. (Note that the difference has the opposite sign to the TIO code so I've swapped the two list elements to compensate.)