For an algebra $$ {\mathbbm{B}}_0:= \left\{{c}_1e+{c}_2\omega :{c}_k\in \mathbb{C},k=1,2\right\} $$ , e2 = ω2 = e, eω = ωe = ω, over the field of complex numbers ℂ, we consider arbitrary bases (e, e2) such that $$ e+2p{e}_2^2+{e}_2^4=0 $$ for any fixed p > 1. We study $$ {\mathbbm{B}}_0 $$ -valued “analytic” functions $$ \Phi \left( xe+y{e}_2\right)={U}_1\left(x,y\right)e+{U}_2\left(x,y\right) ie+{U}_3\left(x,y\right){e}_2+{U}_4\left(x,y\right)i{e}_2 $$ such that their real-valued components Uk, $$ k=\overline{1,4} $$ , satisfy the equation for the stress function u in the case of orthotropic plane deformations $$ \left(\frac{\partial^4}{\partial {x}^4}+2p\frac{\partial^4}{\partial {x}^2\partial {y}^2}+\frac{\partial^4}{\partial {y}^4}\right)u\left(x,y\right)=0 $$ , where x and y are real variables. All functions Φ for which U1 ≡ u are described in the case of a simply connected domain. Particular solutions of the equilibrium system of equations in displacements are found in the form of linear combinations of the components Uk, $$ k=\overline{1,4} $$ , of the function Φ for some plane orthotropic media.