Jelly, 13 bytes

r²ÆṪ=€ḢTḢƲa+¥ 

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A monadic link taking an integer and returning the next integer with shared totient or zero.

Explanation

Main link (takes integer argument z)

r²  | Range from z to z ** 2 inclusive ÆṪ  | Totient function of each   Ʋ | Following as a monad =€Ḣ  | - Check whether each equal to the first, popping the first before doing so T  | - Truthy indices   Ḣ | - Head (returns 0 if the previous link returned an empty list)   ¥ | Following as a dyad with z as the right argument and result of previous step as left (called x below) a | - x and:  + | - (x + z) 

Jelly, 13 12 bytes

r²ÆṪ=€Ḣ$T+⁸Ḣ 

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A monadic link taking an integer and returning the next integer with shared totient or zero.

Explanation

Main link (takes integer argument z)

r² | Range from z to z ** 2 inclusive ÆṪ | Totient function of each $ | Following as a monad =€Ḣ | - Check whether each equal to the first, popping the first before doing so T | Truthy indices +⁸ | Plus z Ḣ | - Head (returns 0 if the previous link yielded an empty list) 

Jelly, 13 bytes

r²ÆṪ=€ḢTḢƲa+¥ 

Try it online!

A monadic link taking an integer and returning the next integer with shared totient or zero.

Jelly, 13 bytes

r²ÆṪ=€ḢTḢƲa+¥ 

Try it online!

A monadic link taking an integer and returning the next integer with shared totient or zero.

Explanation

Main link (takes integer argument z)

r² | Range from z to z ** 2 inclusive ÆṪ | Totient function of each Ʋ | Following as a monad =€Ḣ | - Check whether each equal to the first, popping the first before doing so T | - Truthy indices Ḣ | - Head (returns 0 if the previous link returned an empty list) ¥ | Following as a dyad with z as the right argument and result of previous step as left (called x below) a | - x and: + | - (x + z)