Velocity gradient of surface ocean current in dephts
Yes, the speed of currents varies with depth.

In the case of shallow, coastal water, the flow is retarted by friction at the seabed, so it is slowest at the seabed and usually has a curve of increasing speed towards the surface.

This curve can theoretically be calculated, but in practice it is very sensitive to things like the material and shape of the seabed, wind and waves near the surface, and so forth, so while estimates can be made from theory, it's usually better to measure it. 

Approximations are often made using a "power law" such as 

$u_1 = u_2 \left( \frac{z_2}{z_1} \right)^{\frac{1}{\alpha}}$

where $u_1$ and $u_2$ are the speeds at distances $z_1$ and $z_2$ above the seabed respectively, and $\alpha$ is an arbitrary value often between 5 and 7.
This figure, from Lewis et al. 2017 (creative commons licensed), illustrates a number of such theoretical curves with different values of $\alpha$.



In deeper oceans this mechanism will only apply close to the seabed; however, far more complex systems can come into play, with different water bodies moving in (potentially) different directions. Possibly others with more expertise in ocean dynamics can address that case.