coherent states in a quaternionic setting have been constructed using other methods in [@AliBhatRoy] and [@Thi2]. In this paper we study the possibility of constructing some analogues of the so-called [*non-linear*]{} coherent states on quaternionic Hilbert spaces, using the recently developed holomorphic function theory for quaternionic variables [@Am; @Gra1; @Gra2].

Recall that the real Hermite polynomials are defined by $$\label{I1}
H_n(x)=(-1)^ne^{x^2}\frac{d^n}{dx^n}e^{-x^2}$$ and it is well known that the functions $C_ne^{-x^2/2}H_n(x)$, for some normalization constants $C_n$, are the eigensolutions of the quantum harmonic oscillator [@AFG]. As an immediate extension of the real Hermite polynomials, by replacing the real variable $x$ by a complex number $z$, the complex Hermite polynomials, $H_n(z)$, were studied in [@VE]. In particular, in [@VE], it has been shown that these complex Hermite polynomials form an orthonormal basis of a certain Hilbert space of complex functions over $\mathbb C$ and this Hilbert space is a reproducing kernel Hilbert space. In [@Ga], using these holomorphic Hermite polynomials, a set of coherent states (CS) have been built, which is then used to study some quantum mechanical issues and a quantization of the non-commutative plane. Apart from $H_n(z)$, another interesting set of Hermite polynomials, $H_{n,m}(z,\overline{z})$, were studied in [@Ghan; @Mat; @Wil] 