Think of it this way: Are you tall (height) or are you long (depth)? It depends on if you are standing or laying down. Height and length are just customary linguistic conventions. Two ways of describing the same thing.
How about a cube? It has the same height, width, and depth. Let's say you wanted to increase the cube's width without changing it's depth. Which edges would need to lengthen? It's tempting to say, "Those running perpendicular to my line of sight." But then, somebody standing near you and facing a different side of the cube would disagree. They'd say that by lengthening those sides, you were making it deeper, not wider. Somebody above you facing down might say you are increasing the height. They are all equivalent.
If time is a dimension, then it is just as equivalent.
Let's look at it like Flatland does:
Imagine a point. It exists in three dimensional space, though it has zero dimensions of it's own. But, it is still there a moment later, which means it has a temporal component. So, imagine if you will, a hypothetical vantage point from outside of time. From here, you can see the line that the point has made as it sweeps through the fourth dimension we are calling time. And this line has a length. Call it duration. If the point was there for two minutes, then the line is two minutes long. From this special vantage point, that's just another length measurement. From here, duration and length are the same thing when talking about a line.
Let's add a dimension to our point in three dimensions. We grant it width. It looks to us, here in three dimensions, like a line. But we go to our special vantage point and see that it makes a rectangle as the line sweeps across the fourth dimension we call time. It has length and duration. Or is it width and length? They are the same.
Let's add another dimension: We bend the line around into a circle. Now it has height and width (but, being a circle, has no depth). When we see it from our special vantage point from outside time, however, we see that as the circle sweeps through the fourth dimension it forms a cylinder thanks to it's duration. From here, it can be seen to have height, width, and depth. Which one is duration? It's tempting to say that the straight lines that form the cylinder are duration. But what if it was a square instead of a circle? And let's say it existed just long enough that it's duration was equal to it's width? Now, from the special vantage point, we have a cube - and it's duration is indistinguishable from it's height and width.
So far, we've talked only about objects that self-contain 3 or fewer dimensions including time. But what happens when you talk about an object that is three dimensional in our 3d space (let's say, a cube), but then look at it from our super-temporal vantage point? Now you have a mind-bending shape. It's a cube blurred out along the temporal dimension. It is a true four dimensional object.
From our special vantage point, we can see all four of it's dimensions, and they are all equal in priority because duration is just another length to us here.
Now to blow the mind a little more...
Tachyons, if they exist, are said to exist in "imaginary time". That's not to say that this time exists in our imagination, but rather that it is imaginary in the same way as the imaginary part of a complex number. If you are familiar with imaginary numbers, then you know that they are plotted OFF the "real" number line. They have their own number line axis, running perpendicular to real numbers, forming a plane on which complex numbers can be mapped.
In much the same way, tachyons (again, if they exist at all) exist in imaginary time - time that is somehow "sideways" to the timeline we are used to. This implies a plane on which these tachyons could be mapped.
So, imagining time as that plane, we would then have five dimensions. Our special vantage point, then, would allow us to take that four dimensional cube and 'rotate' it through all five dimensions. If that cube existed just long enough that it's width, height, depth, and duration were all the same and we rotated it within five dimensions, then each of those four dimensions would be indistinguishable from one another.
You could take it another step. What if you are "standing" on that plane of time and look "up"? Now you're talking about a sixth dimension. Three dimensions of space and three of time. And you take that four dimensional cube and roll it sideways to time so it moves through the fifth dimension. In six dimensional space, that movement will create another "line" whose paintbrush is the cube.
And this goes on, ad infinitum, for as many dimensions as actually exist. But not a single one of those dimensions has any property that makes it more or less "important". They all coexist and the universe exists within them.
Just because none is more important than the others doesn't mean that one can't be more mysterious than the others, though. Time, even if it is only a single dimension instead of the mind bending three dimensions described above, is still plenty mysterious. Why is it that we are inexorably drawn along it in one direction but are only able to see in the other direction? It's not anything to do with time itself, but rather how we experience the world around us. To go any further in this vein would remove this topic from Physics and put it into Not Quite Science, so I'll stop that line of reasoning there.
Another way to answer your question (and a much simpler, but -to me- less satisfactory way) would be to point out that we've never numbered the first three dimensions. They are just the "three dimensions" that we are used to. Nobody ever says that a board needs to be cut in the second dimension - they say it needs to be narrower or shorter. First, second, and third have no meaning. So to call time a fourth dimension is simply to acknowledge that we are used to there being only three and now we see that there are four. So it's the fourth one. Nothing special there.
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