We study the following question: "Let $f: \mathbb{C}\to \mathbb{C}$ be an entire function of bounded $l$-index, $\Phi: \mathbb{C}^n\to \mathbb{C}$ be a slice entire function, $n\geq2,$ $l:\mathbb{C}\to \mathbb{R}_+$ be a continuous function. What is a positive continuous function $L:\mathbb{C}^n\to \mathbb{R}_+$ and a direction $\mathbf{b}\in\mathbb{C}^n\setminus\{\mathbf{0}\}$ such that the composite function $f(\Phi(z))$ has bounded $L$-index in the direction $\mathbf{b}$?". In the present paper, early known results on boundedness $L$-index in direction for the composition of entire functions $f(\Phi(z))$ are generalized to the case where $\Phi: \mathbb{C}^n\to \mathbb{C}$ is a slice entire function, i.e. it is an entire function on a complex line $\{z^0+t\mathbf{b}: t\in\mathbb{C}\}$ for any $z^0\in\mathbb{C}^n$ and for a given direction $\mathbf{b}\in\mathbb{C}^n\setminus\{\mathbf{0}\}$. These slice entire functions are not joint holomorphic in the general case. For example, it  allows consideration of functions which are holomorphic in variable $z_1$ and continuous in variable $z_2$.

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