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It seems like the simplest way is to use RuleDelayed twice:

HornerForm[ E^(I x) + 2 E^(2 I x) + 3 E^(3 I x) + 4 E^(4 I x) + 5 E^(5 I x) 
           + 6 E^(6 I x) + 7 E^(7 I x) + 8 E^(8 I x) + 9 E^(9 I x) + 10 E^(10 I x) /. 
E^(Complex[0, b_] x) :> z^b, z] /. z :> E^(I x)

% // TraditionalForm

We couldn't make it simpler since imaginary factors are represented as e.g. Complex[0, 5], not Times[ 5, I] even though the latter automatically evaluates to Complex[0, 5] being an atom:

AtomQ[ Complex[0, 5]]

True

The most straightforward way appears to be using carefully simple replacement rules involving RuleDelayed rather than Rule:

HornerForm[ E^(I x) + 2 E^(2 I x) + 3 E^(3 I x) + 4 E^(4 I x) + 5 E^(5 I x) 
           + 6 E^(6 I x) + 7 E^(7 I x) + 8 E^(8 I x) + 9 E^(9 I x) + 10 E^(10 I x) /. 
E^(Complex[0, b_] x) :> z^b, z] /. z :> E^(I x)

% // TraditionalForm

We should remember one subtlety using patterns in replacement rules involving complex (imaginary) factors, which we can illustrate with e.g.:

FullForm @ Unevaluated[ 5 I x]
FullForm[ 5 I x]

 Unevaluated[ Times[5, I, x]]

 Times[ Complex[0, 5], x]
 

namely: built-in rewriting rules of the system automatically evaluate Times[ 5, I] to Complex[0, 5] being an atom:

AtomQ[ Complex[0, 5]]

True

therefore we couldn't make our rule simpler and had used E^(Complex[0, b_] x) :> z^b instead of something like apparently simpler E^(I b_ x) :> z^b.

It seems like the simplest way:

HornerForm[ E^(I x) + 2 E^(2 I x) + 3 E^(3 I x) + 4 E^(4 I x) + 5 E^(5 I x) 
           + 6 E^(6 I x) + 7 E^(7 I x) + 8 E^(8 I x) + 9 E^(9 I x) + 10 E^(10 I x) /. 
E^(Complex[0, b_] x) :> z^b, z] /. z :> E^(I x)

We couldn't make it simpler since imaginary factors are represented as e.g. Complex[0, 5], not Times[ 5, I] even though the latter automatically evaluates to an atom:

AtomQ[Complex[0, 5]]

True

It seems like the simplest way is to use RuleDelayed twice:

HornerForm[ E^(I x) + 2 E^(2 I x) + 3 E^(3 I x) + 4 E^(4 I x) + 5 E^(5 I x) 
           + 6 E^(6 I x) + 7 E^(7 I x) + 8 E^(8 I x) + 9 E^(9 I x) + 10 E^(10 I x) /. 
E^(Complex[0, b_] x) :> z^b, z] /. z :> E^(I x)

% // TraditionalForm

We couldn't make it simpler since imaginary factors are represented as e.g. Complex[0, 5], not Times[ 5, I] even though the latter automatically evaluates to Complex[0, 5] being an atom:

AtomQ[ Complex[0, 5]]

True

We can make it simply this way:

HornerForm[ E^(I x) + 2 E^(2 I x) + 3 E^(3 I x) + 4 E^(4 I x) + 5 E^(5 I x) 
           + 6 E^(6 I x) + 7 E^(7 I x) + 8 E^(8 I x) + 9 E^(9 I x) + 10 E^(10 I x) /. 
{Power[E, Times[Complex[0, b_], x]] :> z^b}, z] /. z :> E^(I x)

It seems like the simplest way:

HornerForm[ E^(I x) + 2 E^(2 I x) + 3 E^(3 I x) + 4 E^(4 I x) + 5 E^(5 I x) 
           + 6 E^(6 I x) + 7 E^(7 I x) + 8 E^(8 I x) + 9 E^(9 I x) + 10 E^(10 I x) /. 
E^(Complex[0, b_] x) :> z^b, z] /. z :> E^(I x)

We couldn't make it simpler since imaginary factors are represented as e.g. Complex[0, 5], not Times[ 5, I] even though the latter automatically evaluates to an atom:

AtomQ[Complex[0, 5]]

True