There is a known identity $$ \eta\bigg(z+\frac{1}{2}\bigg) = e^{i\pi/24}\,\frac{\eta(2z)^3}{\eta(z)\,\eta(4z)}\,,$$ where $\eta(z) = q^{1/24}\,\prod_{n\geq 1} (1-q^n)$ is the Dedekind eta function, with $q = e^{2\pi i z}$.
My question is if some simpler connection between $\eta(z+1/2)$ and $\eta(z)$ is known?
If you mean algebraic connection, then no. That is, any algebraic equation between $\,\eta(z)\,$ and $\,\eta(z+1/2)\,$ must be homogeneous and therefore the quotient would satisfy an algebraic polynomial equation and hence be constant. The $\,\eta\,$ function is a modular form and there are many theorems from modular form theory that constrain its properties.
The simplest connection I know is: $\, \eta(z)^8 + 16\eta(4z)^8 = (\eta(z+1/2)/e^{i\pi/24})^8. \,$