In this paper we introduce the class of $\mathbb R$-complex Finsler spaces with $(\alpha,\beta)$-metrics and study some important exemples: $\mathbb R$-complex Randers spaces, $\mathbb R$-complex Kropina spaces. The metric tensor field of a $\mathbb R$-complex Finsler space with $(\alpha,\beta)$-metric is determined (\S2). A special approach is dedicated to the $\mathbb R$-complex Randers spaces (\S3).