  In this paper, we study the additive complexity $\rho^{+}_{\mathbf{t}}(n)$ of
a Thue-Morse like sequence $\mathbf{t}=\sigma^{\infty}(0)$ with the morphism
$\sigma: 0\to 01, 1\to 12, 2\to 20$. We show that
$\rho^{+}_{\mathbf{t}}(n)=2\lfloor\log_2(n)\rfloor+3$ for all integers $n\geq
1$. Consequently, $(\rho_{\mathbf{t}}(n))_{n\geq 1}$ is a $2$-regular sequence.
