First of all, a Brownian motion can be indexed by a quite arbitrary parameter set (so-called set-indexed Brownian motion).

Now to your question. There is a natural extension to two-parameters of the ordinary Brownian motion on $\mathbb{R}_+$, namely the Brownian sheet, commonly denoted by $W$ and indexed by $\mathbb{R}_ + ^2 = \{ (s,t):s \ge 0,t \ge 0\} $. A very good textbook on this topic (in a general setting) is "An Introduction to Continuity, Extrema, and Related Topics for General Gaussian Processes"; the relevant pages (6-7) are available online. Now, restricting a Brownian sheet to parametrized curves in $\mathbb{R}^2_+$ results in one-parameter processes, which are continuous centered Gaussian processes. A prominent example is the one-parameter process $A = \{A_t: 0 \leq t \leq 1\}$ defined by $A_t = W_{t,1-t}$. The process $A$ is in fact a Brownian bridge. So, this is an example of a Brownian bridge which is "a kind of Brownian motion indexed by points on" the segment connecting $(0,1)$ and $(1,0)$. However, this is essentially the only way to get a Brownian bridge by restricting a Brownian sheet to some parametrized curve in $\mathbb{R}^2_+$. Nevertheless, a Brownian bridge which is "a kind of Brownian motion indexed by points on the circle" might be obtained as follows. The Brownian sheet is defined, for $s,t \geq 0$, by $W_{s,t} = W((0,s] \times (0,t])$, where $W(A)$, $A \in \mathcal{B}(\mathbb{R}_ + ^2 )$ is the Gaussian white noise based on Lebesgue measure; see pages 6-7 in the aforementioned book. Now, letting the underlying measure $\nu$ in $(\mathbb{R}_ + ^2,\mathcal{B}(\mathbb{R}_ + ^2 ),\nu)$ be a ($\sigma$-finite) measure different from Lebesgue measure, the process $\tilde W$ defined by $\tilde W_{s,t} = W((0,s] \times (0,t])$, where $W$ is a Gaussian white noise based on $\nu$, is no longer a Brownian sheet; it is very likely that one can find $\nu$ such that the process $\tilde A = \{\tilde A_t: 0 \leq t \leq \pi/2\}$ defined by $\tilde A_t = \tilde W_{\sin t, \cos t}$ is a Brownian bridge (which is "a kind of Brownian motion indexed by points on the circle"). I can guide you how to find the suitable $\nu$, if you wish.