  We define higher semiadditive algebraic K-theory, a variant of algebraic
K-theory that takes into account higher semiadditive structure, as enjoyed for
example by the $K(n)$- and $T(n)$-local categories. We prove that it satisfies
a form of the redshift conjecture. Namely, that if $R$ is a ring spectrum of
height $\leq n$, then its semiadditive K-theory is of height $\leq n+1$. Under
further hypothesis on $R$, which are satisfied for example by the Lubin-Tate
spectrum $E_n$, we show that its semiadditive algebraic K-theory is of height
exactly $n+1$. Finally, we connect semiadditive K-theory to $T(n+1)$-localized
K-theory, showing that they coincide for any $p$-invertible ring spectrum and
for the completed Johnson-Wilson spectrum $\widehat{E(n)}$.
